Unlike how bacterial infections could gain or incrementally increase some of its antibiotic immunity or other attributes thanks to the conditions in a single host, for example incomplete or unnecessary antibiotic regimens, viral infections need to be infecting new cells in order to experience meaningful mutations. This is where immune function influences viral mutation.
Less immunity will result in more cell infection which directly creates opportunities for mutations as the virus replicates. If a vaccine improves host immunity enough that they replicate less virus, they will cause much less mutation. The relationship is non-linear because of the multiplicative nature of viral spread (r0?)
My opinion is that isn't anybody's "fault" that a pandemic virus is causing such unbelievable mayhem, but vaccinations and immunology is the truest course to achieving excellent results going forward.
Edit: Virus does not require fresh host to mutate