Current vaccine skeptic thinking is not that vaccines cause autism, it's that they do all sorts of other random and spurious damage to development. Speaking from the front lines, the movement is largely dying out, even in the hotbeds. The Disneyland outbreak tested a lot of peoples' mettle and when it came down to "do I stand on principle" or "do I want my kid to not get measles" people went and vaccinated.
They do come from a place of fear, but they also come from a place of bigotry and moral superiority. Someone who says "I've heard that maybe vaccines cause autism, I'm worried about my kids" usually runs across a "skeptic" who lays down the basix of "you're a fucking idiot, science is proven, how dare you question 200 years of medical wisdom, how dare you endanger herd immunity, this isn't just about you it's about all of humanity and if you don't toe the fucking line you're literally Hitler." At this point the person who was a little worried about vaccines has now decided that the vaccine tribe is filled with assholes and assholes are often wrong.
There's this drive amongst the atheist/skeptic/rationalist community that feels the need to clobber anyone even vaguely outside of their dogma. They're their own worst enemies. If someone comes to you with some doubt about things you know to be true, your job is to walk them through your understanding such that it becomes theirs. yet for the past 15 years the "skeptics" have been on a "burn the witch" rampage that mostly just shows what a bunch of dicks they are.
And I say that with an autographed Shermer on the bookshelf.
For example, anti-vaxxers' ignorance of medicine and scientific process lets people exploit the natural desire for their child to be healthy (or fear that they won't be healthy).
See, you're doing it. Someone who may be just a little bit less certain in their germ theory is guilty of "ignorance of medicine and scientific process." Meanwhile, you use phrases like "exploit" as if the anti-vax crowd were some sort of power-grabbing cult, rather than an insular group of confused and embattled moms who are sick of being shouted at every time they express doubt.
"Doubt" and "skepticism" are synonyms. If someone is "skeptical" of vaccines, the job of the "skeptic" is to answer their doubts ON THEIR TERMS so that they learn. But instead, "skeptics" shake their heads, conclude that anyone who doesn't believe exactly what they do must be "ignorant" and goes online and derides those idiots who endanger their community.