To summarize a huge topic, there are a few different ways to create a working vaccine. One of the polio vaccines is a (usually) harmless variant of the "live" virus; the other is from the inactivated or "dead" virus. With a live virus, there is the possibility of mutation, so I think that technique has fallen out of favor.

The mRNA COVID-19 vaccines used in the US do not include any part of the virus; they simply provide the "blueprint" for making one of the viral proteins. The immune system uses the blueprint to learn how to attack anything that carries that protein, i.e., the COVID virus. So there is nothing alive that can mutate in the vaccines themselves.