On one side we have people who have chosen to believe something that is in direct conflict with all facts and observable reality, and on the other hand we have people who are attacked for trying to explain the situation rationally. I'm not saying that
you're attacking them, but the attacks are commonplace and widespread.
The more the medical community tries to marginalize people who have those questions, the more these folks will distrust what science is telling them.
The more the media tries to normalize people who declare that vaccines are the cause of autism, the more people will feel justified in making decisions based wholly upon ignorance and fear. The fault here does not originate from the people attempting to combat fear with fact.
So in this case I think it is wise to encompass these people, say yes we understand why you would wonder about that, here is what the studies are showing and more are being done. We hope to someday understand why this happened to your child.
That would be true if we were working from a level playing field, but we are not. An entire industry is devoted to preemptively convincing parents that vaccines are unnecessary and dangerous, stoking their fear and rendering them incapable of discussing the matter rationally.
Therefore it is unacceptable for you to require that science abandon facts and reason when explaining the reality of the situation. Rather than acknowledging the people who actually have the answers, you're telling them to accommodate the people who propagate (and profit from) the lies.
Incidentally, I've had this conversation with many people who believe that vaccines cause autism, and I have found that it is absolutely futile to "understand" or "empathize with" them. In all cases, they walk away from the discussion even more thoroughly convinced that vaccines cause autism, in part because
a person who believes in science says he understands their fear. They wholly disregard the facts that I might present, and they simply cling to their fear. But if I express my hostility to the notion of a vaccine/autism link, then at least I know that I'm not validating their irrational belief.
Simply refuting them only leads to more distrust. And it is spreading. Thinking vaccines are dangerous now seems to be a popular conspiracy theory among right-wing crazies. Someday it may be as divisive as the abortion debate.
If you think that it's a conspiracy theory among right-wing crazies, then you haven't been paying attention. It's an insane fear that has infected a large fraction of the population. And it needs to be fought at every opportunity.
Extremism - in either direction - forces people to pick a side, and opposes consensus.
It is not extremism to say "here are the facts" and then demonstrate those facts. Extremism arises from declaring something true or false in the absence of (or in spite of) the facts. It is not extremism to say "there is no rational reason to believe that vaccines cause autism."
Ultimately, you are attempting stop the conflict without resolving it, and in so doing you are fostering an atmosphere in which ignorance and fear will drive parents to make dangerous and irrational decisions regarding their children's health.