You don't have to be Muslim to believe this sort of stuff. There was the MMR fuss, and at the moment (mement?) the fundies are pushing conspiracy theories about contraception, abortion and HPV vaccination, the latter with help from the unaffiliated anti-vaccine crowd.

It's very tempting to go with intuition. Instinctively, parents do not want their child to be "harmed", i.e. have a needle stuck in them. They don't want them to be upset or have to deal with the pain of it. They then rationalise that by allowing themselves to be convinced by the anti-vaccine stories. If their child is sick, and that happens around the time of the vaccination, they blame the vaccine. If they hear the stories of such people, they become concerned (and the story tellers are self-selecting: those who don't have problems don't have such passion) and again their instinct is to be protective. And, to be fair, large pharmaceutical and synthetic chemical companies have been pretty badly behaved, both as of late and in the past (valium, oxycodone, thalidomide, DDT, agent orange, etc). The Nixon medical system in the US is an appalling farce of perverse incentives and exploitation. It's not surprising if they don't trust the medical establishment, sadly.