Like everything it starts in the states. Of course infectious disease doesn't give a damn about state borders, but neither does this kind of policy. It took 50 years for them to do it to abortion, but there's no reason to expect it to take that long with vaccines.
As an abortion expert, it's extra scary that the anti-vaccine folks grift off their wins - they sell the supplements and scams that people turn to when they reject vaccination. This profit motive could really supercharge the pace of vaccine restriction.
Sure, some anti-abortion folks make money off their opposition, but as far as I can tell, their money making is neither as scalable nor as directly tied to restrictions as are the supplement/wellness things that vaccine opponents hawk.
I would argue CPCs and abstinence only ed are the supplement/wellness of the anti-abortion industry and they've just managed it better because they go straight to government and churches as the client rather than individuals.