> and since unvaccinated people are more likely to cause the virus to mutate

If you understand natural selection, then this assertion makes zero sense. Where in the genome do we see the accumulation of mutations in all covid variants seen so far? In their S-protein. Where does Omicron have 30+ mutations? In its S-protein.

What is the single protein that the mRNA vaccines code for??

Yes, if you reason from first principles, it is trivial to see that it is in fact the throngs of the vaccinated who are driving the evolution of SARS-CoV-2!