So, I’m not a virologist, so I can’t answer about viruses. But I am a bacterial microbiologist, so I can talk a bit about pathogenic bacteria. Short answer: yes. Long answer: yes, kind of.
It really depends on what the vaccine is targeting and what the pathogen is. My favorite pathogen is Streptococcus pneumoniae, the leading cause of pneumonia. So let’s look at it from that perspective. There are vaccines for S. pneumo, but the vaccines only target certain stains of S. pneumo. And every 5 or so years, we make a new version of the vaccine because the types of S. pneumo that are causing disease keeps shifting. If the vaccine accounts for type A, then type B starts to cause more disease. If the vaccine accounts for types A and B, then type C starts to cause more disease. If the vaccine amounts for types A through C, then type D starts to cause more disease. Repeat ad nauseum.
So yes, we can cause shifts in pathogen populations through vaccines. This is evolution, in its strictest definition. That being said, there’s a lot of caveats. First, that doesn’t mean that vaccines are bad. Maybe we want to shift the population (for instance, toward a milder form of the disease). Or maybe it doesn’t strictly matter if the shift occurs (if we can just keep making new vaccine versions, a la S. pneumo).
Second, even though vaccines may be shifting the population, that doesn’t mean that it doesn’t work. The S. pneumo vaccine significantly decreased infection and mortality from pneumonia. And while a lot of people still die from pneumonia today, it’s nothing compared to the mortality before modern medical science.
Third, it really depends on the vaccine. Specifically, how hard is it for the pathogen to mutate that portion that the vaccine is attempting to mimic? There are certain proteins that are more difficult to mutate than others. For instance, there are certain proteins that are involved in binding to and attacking the host. These proteins tend to be somewhat difficult to mutate, since mutating those proteins tend to also make the pathogen less efficient at attacking the host. If the vaccine trains the immune system to recognize these proteins, it can be really difficult for pathogens to evolve away from these proteins. Not to say that it’s impossible for pathogens to evolve anyways (pathogens are surprisingly tricky), but a well-designed vaccine, along with good adoption in the population, can significantly hinder a disease.