But why do they believe it?
I think they believe it because it makes them feel good.
This is important because if they believed it because they'd looked at the science etc, it would mean they could be convinced by more science. But if its because of feeling, no amount of pro-vax evidence would convince them.
That's the point here: they'll die for their beliefs (some of them at least). But why do they believe it?
Until you know why, you can't challenge the belief. Your only option is to challenge the behaviour, and that's where things get messy...
What you find when you look into it is that yes vaccines are safe but only because we take stringent measures to ensure that safety. They are not safe by default as most people believe. Then what you have is Anti Vax people believing they have some untapped knowledge because when they talk to people about it very few know exactly why or how vaccines are safe, just that they are which coming from the Anti Vax perspective makes you feel you know something other people don't which is enough to validate your opinion.
This doesn't take into account risk reward even if they were not safe but that's another story.
That's why you recognize that it would not go well to attack this with physical force - the strength of people's reaction would not because they want to appear special, but because they honestly, 100% believe someone is trying to hurt their children.