The only possible side effect I've read about so far was temporary Bell's Palsy, and that was limited to the Pfizer vaccine and unconfirmed to be caused by the vaccine. Anti-vax nonsense is not acceptable - if there are any scientifically proven side effects then such claims must be substantiated with relevant studies.
Or coincidence, since the number is incredibly low. Claims like this must be substantiated, otherwise you're just guessing. If I take the vaccine and have a car accident later, that's not the vaccine's fault. You have to consider externalities, a lot of the patients taking it are already old.