Common accepted knowledge before 2020 was that yes, the vaccines cause mutations / variants as the virus attempts to "work around" the impediment and continue survival. Plus, factoriing in that these aren't vaccines, only gene therapy to force your body to defend against one spike protein and not the virus as an entire entity, and it is suddenly easy for the virus to adapt and circumvent.
The notion that the unvaccinated are causing the proliferation of the variants is insane and laughable. But, welcome to Current Year.