Ideology can't be fought with arguments. As the saying goes, you can't reason someone out of a position they didn't reason themselves into. People don't have ideologies because they convinced themselves of them with a reasoned argument. And that is not by any means limited to ideologies like "anti-vax" (a better example would be Flat Earthers since there actually are reasonable arguments to be made against requiring universal vaccination for at least some vaccines). We all have ideologies, and some, perhaps many, of them are daft and will end up being considered daft by future generations.
That is the reason why you can't suppress "unwanted" speech--because nobody has a direct line to "the truth" and nobody has the ability (let alone the right) to sit in judgment on people's expression of their opinions and beliefs, no matter what those opinions and beliefs are based on.
Anti-vax ideology needs to be fought with better arguments, not with sheer suppression, which will only drive some distrustful people towards the "hey, why are THEY trying to ban this information" position.
At the end of the day, the problem is one of trust, and you don't gain any trust by treating other people's opinions heavy-handedly.