Below is from the CDC (first reference I checked and how I understand it):
"What You Need to Know
- New variants of the virus are expected to occur.
- Slowing the spread of the virus, by protecting yourself and others, can help slow the emergence of new variants."
Below is from the University of Maryland Medical System (second reference I checked):
"The vaccines help stop new variants.
As we vaccinate and boost as many people as possible, as quickly as possible, we can help stop the
spread of the coronavirus. A decrease in transmission of the virus means fewer opportunities for it to mutate, which can help prevent the emergence of any other variants."
I'm not aware of info that says getting vaccinated makes the virus mutate, period. Not that there isn't some logic to it like overusing sanitizer or taking too many antibiotics which can make those two methods ineffective.
Not a scientist but I would then think things like smallpox and polio would still be quite active since they would have mutated due to the smallpox vaccine, for example.