@dittheringdoldrums
There's no clinical evidence of any long-term impact on fertility and no plausible mechanism by which the vaccines could have such an effect.
I think the reason for the concerns about this is that a small number of women have reported disruption of their menstrual cycle in the few months following vaccination. This isn't a new phenomenon - it can occur with many vaccines and with many situations in which people are mounting an immune response to something. The basic explanation is that the body is focusing on the immune response, so things like reproductive cycles don't have the same priority for a little bit - once the body has figured out the immune response, normal service is resumed. This can also happen with flu vaccines, vaccines for travelling abroad, mild infections, etc... It can be frustrating for people TTC but it's not a lasting obstacle. Most people don't experience it at all, and people who do tend to find their cycles are back to normal in a couple of months.
COVID, however, does have effects on fertility for men and women, and can linger in tissues for a long time (recent data suggests it can still be detected after a year!). One of the receptors it uses to gain entry to cells is widely expressed in reproductive tissues. COVID infection can cause temporarily reduced production of functional sperm, temporary reduction in AMH level (a marker of ovarian function), etc. While these effects are also considered likely to be temporary, they can be much longer-lasting and there is much less data about how they resolve.
Basically, vaccines = no evidence of lasting effects on fertility although a small number of women may have altered cycles for a couple of months; COVID = evidence of effects on fertility which seem to resolve in time but we need more data to know the longer-term effects.