Worries about COVID vaccines and infertility are ridiculous
Public Service Announcement: Claims that the COVID vaccine could cause infertility are somewhere between misguided and malicious. There is no more reason to think that COVID vaccines can cause infertility than that they could cause diabetes, or heart attacks, or arthritis, or irritable bowel sydrome, or anything else.
Vaccines are not entirely irrelevant to pregnancy. As a general rule, pregnant women are advised against vaccines that consist of a live attenuated virus or bacteria, out of abundance of caution. The the MMR (measles, mumps, rubella) vaccine stands out as one example. However, women are sometimes vaccinated before they know that they are pregnant and, fortunately, adverse effects on the fetus have not been demonstrated. Thus, the risk to pregnancy seems to be more theoretical than real. (And just for clarity, COVID vaccines do not contain live attenuated virus.)
Pregnancy aside, in the history of vaccination there have been a few vaccines that were pulled from the market due to safety problems. In the 1960s, an RSV vaccine for children was found to actually exacerbate disease [1]. In the late 1990s a vaccine against rotavirus was pulled after it was associated with a small risk of intestinal intussusception (where a segment of the intestine folds back on itself). Other vaccines have special safety considerations. The live polio vaccine, for example, is no longer used in countries with low polio risk because the virus can sometimes revert to virulence due to mutation in its genome [2]. As another example, the smallpox vaccine is still used in certain laboratory situations even though smallpox has been eradicated [3]. However, skin conditions like eczema and psoriasis pose a safety risk for this particular vaccine. The kinds of issues that I have described are well known among vaccine researchers and clinicians.
It is one thing to worry about how a live vaccine might theoretically affect a fetus, which doesn't have a functioning immune system. Similarly, I can understand being cautious when there are no safety data on pregnancy. But I cannot think of any vaccine used in the U.S. or internationally that is even suspected of causing infertility. It just isn't a thing.
There certainly are infectious diseases that can cause infertility, which leads me to a question: Why would you be more worried about a vaccine causing infertility than about the disease itself? It makes no sense. But that's probably because it isn't supposed to make sense; it's just supposed to scare.
Notes:
1. That a vaccine could make a disease worse is counter-intuitive at first glance, but there are exceptions to most rules in biology.
2. The live polio vaccine is superior to the killed vaccine in several respects, which is why it is still used in certain geographies where health infrastructure is poor.
3. It's not because they are doing smallpox research. It's because the virus that was used as the smallpox vaccine--called vaccinia--is a useful laboratory tool. Inadvertant infections can be nasty, so lab workers may be vaccinated as a precaution.

