It's probably reasonable to say that the anti-vax movement started out on "The Left" though mostly that particular West Coast slice of "The Left" which is much more about attitude and personal lifestyle choices than actual politics, but since then it's metastasized and is largely a right wing nutter thing. I don't quite get it, though. Trying to read some of The Discourse among its adherents doesn't leave me very enlightened. It doesn't seem to be much deeper than vaccines are bad, and we special people know that, and therefore do not vaccinate our kids, and people who do vaccinate their kids are bad because reasons.
Yes once upon a time the supposed vaccine-autism link was the big thing...now I'm not even sure? It's still there, but it isn't the only thing.