What's important to note, though, is that the mutations are blind to their environment. The virus or bacteria doesn't see an antibiotic or antibody and say "I need to adjust this aspect." It's just constantly changing as it reproduces, and some of them may end up winning a battle that the original did not. And then they spread from there as would any strain.

From the purposes of the virus, an unvaccinated person in a poor country without access to vaccines isn't better or worse than one in the US who's simply refusing. It doesn't know. They're both opportunities for the virus to keep replication and so keep (randomly, blindly) mutating.

If the odds of reproducing in a way that escapes people's current antibodies, and making your way outside the host (because a mutation the immune system adapts to before it can be spread dies there) are 1 in ten billion, say... and you leave 7 billion people in the world unvaccinated... suddenly those odds aren't so great. (Numbers entirely fake here, but basically, less likely events become more likely as more time goes by, if you keep giving them opportunities to happen.)

A key advantage of a vaccine over an antibiotic is that it can potentially stops the problem before it starts whereas antibiotics are generally given if you already have a nasty infection. So the amount of virus that's exposed to the immune system in a vaccinated person will be lower, and throttled more quickly, giving less time for it to get lucky, IIRC. But nothing about that would magically necessarily stop a virus that's still spreading like wildfire in a large unvaccinated population from coincidentally developing new traits that help it out. The goal is to vaccinate a large enough population quickly enough that this "spreading like wildfire" situation stops, so we reduce the mutation rate of the virus.