Is there any evidence pointing in that direction?
I am not a biologist, but I have a basic understanding of how the mRNA vaccines work. As far as I know the protein the vaccine codes for is well understood.
The only kind of sketch part to me is whatever complicated bio-chem is used to enable the uptake of the mRNA, but from what I've read it seems like that process is a one time thing, not some lasting DNA based change.
Which is to say, at an intuitive level, I would expect to see any harsh side-effects immediately (hours to days). Again, just a layman's interpretation, like most of the anti-vax comments on here.