That's not how evolutionary pressure works. If fewer people are infected, the virus has fewer opportunities to mutate. Among those people, any surviving mutation has (by definition) a greater ability to escape the vaccine, but that's not the same thing as saying that the vaccine increases the chance of a new mutation.
There's a missing part to this by the way: you need a reservoir for new mutations to replicate in, and they have to be mutations which become dominant in the reservoir.
Since the vaccines reduced R(eff), generally below 1 (so far observed), any chain of infection through vaccinated people tends to terminate - not continue. Which means however vaccine evading that virus is, all of it dies.
This all changes if you have a large group of unvaccinated people presenting no challenge to it. Freely spreading for a whole lot of cycles through that population means more vaccine-resistant copies are now out there, with more opportunity to challenge vaccine resistant individuals they come into contact with (since R(eff) in the unvaccinated is ~8).
R(eff) lowers the higher the vaccination rate. There are some pretty obvious % targets depending on your R(eff) initial. For the initial population number it's ~90%.[1]
"no reliable evidence" - of course vaccination helps bring R(eff) lower. That's just the way the math works. If R(eff) is already low enough due to other mitigation, then additional vaccination helps bring it below 1. We see that all over the nation now, R(eff) rates dipping below 1 as vaccination rates increase.
I have trouble visualizing that, but if that's true it's just more reason to do whatever we can to get local Rt < 1. Vaccines get us partway there, other mitigation efforts like mask use, distancing (and natural immunity) get us partway there as well.
I'm personally vaccinated and not very worried about getting the disease if I have to be out and about. I feel my personal risk is low. But in terms of societal/community risk to others, that's why I have a personal policy of limiting myself socially so I don't contribute to spread.
The local metrics I personally follow are:
1) Is Rt above 1?
2) Are cases/100k above 10?
3) Is test positivity above 5%?
4) Is ICU usage above 85%
If any of those are on the wrong side, I'm limiting my social gatherings.
