The more people that have a virus, the more copies are out there mutating, and the more mutations (in total) you'll see.

If you introduce negative pressure, like a vaccine or antibiotics, some of those mutations that were previously deleterious will turn out to be a net benefit. E.g: A thicker cell wall in bacteria makes it harder to absorb nutrients, but also increases the dose of antibiotic it can survive. These mutations happen all the time by chance, but in the absence of the vaccine/antibiotics, they are less fit and will tend to die off.

A vaccine that eliminates a virus is likely to result in less infections. Less infections means less mutations means less chance for a dangerous variant. If some variants do evolve that escape the vaccine, odds are good that they will be less dangerous than the original, not more.