Until recently, antivax was largely a liberal form of anti-intellectualism. It was a reaction against large pharmaceutical companies.
It didn't really become a conservative position until COVID. It's mostly an anti-progressive thing, but builds on existing populist conservative anti-science attitudes. (Conservative doesn't always mean anti-science, but populist versions of it will inevitably tend that way.)
The problem is calling it anti-intellectualism implies that one can be conservative in "right" ways. And in modern times, being conservative in "right" ways means you are basically a liberal.
The whole idea of conservatism is that certain people in a population are going to have "non desirable" qualities, and the best thing would be if those people just "went away", but realistically you can't advocate for that, so you have to do mental gymnastics and say "those people should just work to become better, or get punished".