I'm not sure if there's a conscious choice to ignore logic, or if it's solely based on mistrust and paranoia?

Hanlon's razor guides us to

Never attribute to malice that which is adequately explained by stupidity.

Now I don't really think anti-vaxxers are stupid as such, at least, not all of them. I think it's unproductive to make that assertion. Some people might say I'm stupid for believing some of the things I do, but I'd rather they didn't, and I believe in treating others how I'd like to be treated.

I also don't think they're purposefully ignoring logic. Instead, they are driven by paranoia and distrust into a sort of self-perpetuated skepticism of life, the universe, and everything. Just as vaccines give us herd immunity, anti-vaxxers share amongst themselves a sort of herd ignorance. It's involuntary.

We can prove vaccines work. We have data to support it and a widespread, decentralised, and organic web of trust confirms it. But something happened with these folks' lives which shook their faith in that web of trust, or did not allow it to establish in the first place. If you are raised believing not to trust the establishment-- as part of a "chosen people"-type religious sect, for example-- then you won't know any better than to immediately dismiss anything widely regarded as fact simply because it is the mainstream status quo.

Anyone who's ever been religious can understand the cognitive dissonance encountered when considering evidence which contradicts dogma. There's an interesting sort of ever-present nagging feeling that none of this is real. That it's all a test. If you've never been religious, a way to awaken similar feelings within yourself is to consider that you are living inside a simulation. Imagine the skepticism with which you would approach almost every assertion from someone who doesn't share your belief in the simulation. You simply cannot get anywhere in a debate against this sort of reasoning. The only winning move is not to play.