All evidence shows it to be the other way around, the vaccinated are far more dangerous to the unvaccinated than vice versa, it isn't even close. Leaky vaccines create more virulent and deadlier viruses in the unvaccinated.
Mutations and variants occur all the time as the virus reproduces. Variants are not a response to any vaccine; they have appeared primarily (entirely? I haven't been following all of them) in regions where the vaccine is not yet widely available because that is where the virus is able to multiply (and thus mutate) the fastest.
Interesting theory though, I don't think it's true right now, but potentially there may be evidence I'm not aware of.
A real life/modern example of this can be found in Marek's disease in chickens. https://en.wikipedia.org/wiki/Marek%27s_disease#Prevention
The counter claim seems to be that because vaccination reduces the abundance of the virus there is a lower chance of mutations. It's hard to say which effect is stronger and it probably varies greatly by virus and by vaccine!
Disclaimer: I'm not an expert, just an armchair enthusiast.
“Vaccines that let the hosts survive but do not prevent the spread of the pathogen relax this selection, allowing the evolution of hotter pathogens to occur. This type of vaccine is often called a leaky vaccine.”
https://journals.plos.org/plosbiology/article?id=10.1371/jou...