Xtykutl

First, I'll say that it would be a lot better to have your characters descend from or be raised by capital D Deaf people. This way you get an established Sign Language (which can morph like any language will) vs having to spontaneously create one. I've actually seen a first generation Sign Language (wrote my undergraduate thesis on it) and it's not as grammatically complex as an established language (though it for sure had grammar) and the vocabulary wasn't very big either. Plus you have consistency issues. In this case, the language didn't start up until the children were old enough to go to school, and there were strong influences from visitors and of course the native spoken language. While doing it this way can make an interesting story, if you want to honor Deaf culture, preserve an existing language with native users.

There's a terrific short story in this anthology:
https://www.goodreads.com/book/show/23129839-octavia-s-brood

Octavia's Brood: Science Fiction Stories from Social Justice Movements

Earth transports babies deemed unacceptable due to their disabilities to a space colony. Which turns out to be a pretty awesome place.

While of course a lot of deafness is acquired, for those born deaf, even existing technology can diagnosis it (all newborns in hospitals get the screening tests and there are further tests for those who "fail").

Another way to get a Deaf community is to have a colony come from a group of workers or settlers chosen for their deafness. Because it was advantageous for that location or the work. As they raise families, many of the children will also be deaf, since some deafness is genetic (you can even screen for that initially). And everyone will grow up signing.

Depending on what's around, people born hearing might not thrive, or might live with whatever noise there is, or the noise could be gone by then, or they might choose (or their parents will choose) to make them deaf medically for their own comfort and health. I can imagine all the hearing readers gasping right now. But the reality is that our society routinely removes someones ability or even bodypart that is not deemed important, because there are other benefits. Extra fingers? who needs those? A quick clip of the auditory nerve could save a lot of children in this colony.

Your colonists have Waardenburg syndrome.

https://healthjade.com/waardenburg-syndrome/

https://en.wikipedia.org/wiki/Waardenburg_syndrome

Waardenburg syndrome is a rare genetic disorder most often
characterized by varying degrees of deafness, minor defects in
structures arising from the neural crest, and pigmentation changes.

There are a lot of other images of folks with this syndrome on the web. They characteristicaly have a white forelock, bright blue irises and wideset eyes. The Wikipedia article states that about 1 in 30 students at schools for the deaf have this syndrome and so they can be cognitively normal.

For your story, there is some tremendous benefit conferred by Waardenburg syndrome as regards living on your space station, such that all available Waardenburg embryos were chosen to grow up. You will need to invent this benefit or read more about the syndrome than I have. An easy one is that for reasons beyond control, your space station is irrevocably loud such that inhibitants are driven to deafness anyway, and having Waardenburg's is an easy way to ensure deafness without also having more debilitating congenital anomalies. Or you could give them something extra - for example an abnormal accumulation of neural crest cells in the pituitary (the vestigial "third eye" in the center of our brain) give that organ back the powers that it lost during our evolution

Perhaps your colony was designed to be deaf.

This isn't as far-fetched as it may seem at first glance. Today there are many known instances of parents intentionally having children with a “disability”. Using preimplantation genetic diagnosis parents can choose the traits of their children. While generally, we assume that parents would choose to have “healthy” children many parents are using these services to have “disabled” children, generally in the same way they themselves are “disabled”. Your use of Deafness with a capital D suggests you already are aware of this, but many "disabled" communities, Deafness perhaps chief among them, think of deafness as less of a handicap and more of something like a culture. Advocates of this idea feel that being deaf isn’t a significant impediment to their everyday lives and rather enriches their lives in various ways.

New colonies are often formed by those dissatisfied with their current environment or who are being persecuted for one reason or another. Perhaps your Deaf colony arises from a Deaf community fleeing laws outlawing their continued existence or otherwise seeking independence.

You might want widespread congenital cytomegalovirus (CMV) infection.

Most of the medical terminology on a cursory search is well over my head here, but to pick out the dramatic highlights:

Cytomegalovirus (pronounced sy-toe-MEG-a-low-vy-rus), or CMV, is a common virus that infects people of all ages. Over half of adults by age 40 have been infected with CMV. Once CMV is in a person’s body, it stays there for life and can reactivate. Most people infected with CMV show no signs or symptoms. However, CMV infection can cause serious health problems for people with weakened immune systems and for unborn babies (congenital CMV).

And also:

The diagnostic rate using combined genetic deafness test and CMV DNA detection test was determined to be 46.4% in bilateral profound sensorineural hearing loss (SNHL).

At first blush it looks like it has some promise.

2.What sorts of causes might result in a population with widespread generational deafness such as this?

First, in the case of generational colony ships and genetic mutations, this is highly unlikely as both the populace would be prescreened for such defects and likely trained to be selective in their breeding. Mutations could still happen, its just unlikely that it would propagate to the whole population.

But here is a plausible idea:

Over Exposure to Loud Noises

During the generational voyage a component was damaged beyond repair. That component was responsible for dampening sounds generated from [insert loud machine component like the engine]. Thus the populace self mutilated themselves so as to not need to hear so they can sleep. The society then adopted sign language as their official means of communication.

Fast forward to the colonization, after generations of the above situation the populace both forgot how to speak the spoken word and are use to and prefer sign language.

An alternative approach with non-biological causes: The generation ship was prone to catastrophic loss of air pressure. Automated systems and rapid response from the staff meant pressure loss didn't last more than a few seconds each time (and people can survive that, it's just uncomfortable), but the air pressure loss was significant when it occurred. Thanks lowest bidder contracting!

Most of the negative effects of brief exposure to near-vacuum conditions are unpleasant, but not long term. But unfortunately, your sinuses are bad at venting pressure quickly; when the pressure drops rapidly from 14.7 psi to less than 5 psi, your eardrum is almost guaranteed to rupture. Now, any given rupture generally heals within a few months, but unfortunately, the steady drip of air pressure problems means they usually got reruptured over and over before the healing completed, and eventually the scar tissue stiffened the eardrums to the point of deafness.

So after a few decades of this, your skeleton crew arrives at a new planet stone deaf, having long ago discarded spoken language in favor of text and sign communication as they all went permanently deaf.

As it happens, all the embryos have perfectly functional ears. Unfortunately, no one thought to send along recordings of the language, and everyone who knew how to speak it is decades out of practice (or dead, if the original crew died en route to be replaced by children born in space), and can't actually hear what they're saying; the newly decanted babies simply don't have any consistent model for speaking the language. So they learn sign instead. And it works. And reinventing spoken language from scratch is not that easy, and when it happens, each local group does it independently, ending up with mutually unintelligible spoken language, so sign becomes the lingua franca. Even if the occasional group speaks aloud amongst themselves, they have little choice but to fall back on signing (which everyone inherited from the original skeleton crew) when dealing with outsiders.

Add on some additional motivations for signing (e.g. the planet itself has a low pressure atmosphere or constant high winds which reduce how well sound carries and/or require respirators that muffle your voice, or have dangerous critters that attack noisy folks), and the system can end up self-sustaining for quite a while.

The statements

"...a wide variety of frozen/preserved embryos"

and

"those embryos happened to have a high frequency of deafness"

seem to be a statistically very unlikely combination. Any sensible expedition would be sure to have plenty of genetic diversity and that is implied by 'a wide variety'.

However if the embryos were all planted by a mad scientist (probably all his/her own offspring) then massive inbreeding would occur. It would be very likely that a genetic defect such as deafness would arise.

Alternatively, perhaps many of the embryos were destroyed in transit. Again inbreeding would be a serious problem.

If you are looking for a plausible reason behind everyone being able to sign, we need look no further than the vacuum of space. For a space traveling culture a language immune to radio failure and decompression would be invaluable. On a generation ship, it would be taught equal to spoken language, to maximize the odds of survival.

In your universe, people should already feature brain-to-brain, brain-to-machine and machine-to-brain full-duplex efficient communication. This means that communicating through verbal means is something outdated and inneficient. Even typing things is inneficient. Further, in that scenario, many people do not even need to develop or learn spoken languages.

Space is quiet. If you live your whole live in a spaceship, you aren't going to hear anything from the outside. If your spaceship is quiet, there should be no need to hear anything.

Things happening inside the spaceship could be detected by machinery installed into the spaceship itself. If needed, sounds could be captured with microphones and transmitted to humans via a machine-to-brain mechanism. This also have the advantage that frequencies that are way out of the human biologically-hearing ranges can be heard perfectly.

This means that hearing through the biologic hear is not needed anymore. Talking isn't needed either.

Now, let's suppose that the ship is not so quiet. That damn reactor produces a very loud awkward sound! That combustion chambers produces horrible echoes around all the ship! Couldn't we be better if we just choose to not hear them? If those sounds are captured by a biological hear, then we can't filter them out and they bother and hurt too much for being tolerable. So, it would be better to be biologically deaf and hear things only via the machine-to-brain interfaces. The machine is responsible to filter out unpleasant or irrelevant noise.

The other answers are quite detailed, but if you're looking for a quick 'n easy solution, just handwave Cosmic rays. Make 'cosmic deafness' due to space radiation a known occupational hazard for early colonists, before the cure was developed. Even worse, for some areas (colonies in the Oort cloud for example) it's genetic and affects offspring.

Maybe they can hear but not speak?

If your objective is a human society using sign language instead of spoken communication, you may wish to consider the flipside of deafness - a population that can hear but is dumb (in the medical sense). To quote from an article on the Australian ABC News website

Our ancestors, as well as modern apes like chimpanzees, didn't have a
space between the horizontal tube that leads into the jaw and the
vertical tube that leads to your throat, the way modern humans do.

But the space, which makes it possible for us to choke on food, also
allows us to do something else: speak.

Speaking is a huge advantage and choking on food is a low order risk in our current situation on Earth. However, there could be a combination of environmental factors that make the choking risk much greater for a population on an offworld colony somewhere. As a result, the ability to speak was removed through genetic and/or surgical manipulation in order to remove the choking hazard. This may have been intended as a short term solution but gained cultural inertia due to other situations where sign language was advantageous and/or speech was a liability.

As an alternative or addition, if the critical requirement is Deafness rather than sign language use, have some parasitic lifeforms akin to a tick in size that target the human ear as an entry point into the body. The parasites require still air, so they do not target the nose or mouth where respiration is almost constantly occurring, but the ear is ideal for them. As per the above scenario, the ears needed to be sealed in order to remove the entry point by surgical and/or genetic means.

They all work in the engine section of the ship. The permanent high noise means ear protection is worn at all times and people communicate using sign language to the extent that after multiple generations in that environment it's just what they always use.

You don't need deafness to suppress speech. Space is vacuum, space ships/stations are bunch of vessels with different pressures separated by vacuum. Voice communications are unreliable in such environments without intercoms (which could be made unreliable via other means). Eardrum damage due to sudden pressure changes reinforces the drift away from voice toward more reliable means.

You can easily have a population that uses dominant sign/body language because of entire life spent in ship environment. It becomes part of space-dweller culture, retained when raising the settled-down generation. Much like a hearing kid raised among deaf people, they can be capable of vocalizations and hearing, but for them every spoken language would be a foreign language.