This part of the problem is irrelevant - the ammunition of most such people is infinite. They point at everything, and say "look, I told you" about legitimate and illegitimate things alike, and they imply it serves their beliefs, but in reality their beliefs are exactly as unflappable after each "discovery" as before.

Think about the logic of your statement here.

You are starting from the premise that you are correct, and that any evidence against your stance is a problem because it gives the people you have up-front decided are wrong more evidence.

Wouldn't a rational person say "here's some new information, maybe I should incorporate it into my views?"

This can't be a reason to avoid legitimate criticism, though. In fairness, the anti-vaxxers might "produce" some of their own "evidence" regardless of what happens in the real world.

Anti vaxxers don't need ammunition. They're the ones who are spouting absolute crap about turning into a butterfly and 5g mind control. No amount of logic will ever convince them

Vaccine sceptics are the ones who are swayed by this, and are much more logical people. They're the ones who read the information and are still unsure. I can't fault them for making an informed decision, but it's difficult when there's so much conflicting information in such a short amount of time.

They'll never have a greater piece of ammo than the term "anti-vaxxer", which immediately shuts down any discussion involving skepticism of these vaccines.