This is a very common conversation I have on a daily basis. I stress again and again that we have never approved a vaccine that subsequently was shown to have caused delayed side effects years later. The longest delay in severe side effects is about 6 weeks for extremely rare cases of Guillan-Barre syndrome associated with certain vaccines (but not within any of the approved COVID vaccines), and even with this, the onset is usually much sooner. Out of an abundance of caution, all vaccines need at least 3 months of safety data. Beyond that, there really isn't a viable biological mechanism for a vaccine to suddenly cause problems months or years later.
All of the vaccines approved in the US met the required observation period for safety. After hundreds of millions of dosed of Pfizer and Moderna, there have been no surprises. The small chance of venous sinus thrombosis observed with Johnson & Johnson is exactly the reason why post-marketing surveillance continues. The safety record of the mRNA vaccines could not now be more crystal clear.
Finally, even the idea that we don't have long-term safety data on mRNA vaccines is untrue. mRNA vaccines have been under study for other viral diseases, such as rabies, CMV and Zika, for over a decade. These vaccines are not yet marketed, though, because the trials have not so far hit their efficacy endpoints, but their safety is not in question after years of observation.