Autism has been on an incredible rise the past few decades.
1 out of 68 kids in the USA is born with some sort of Autism Spectrum disorder, according to the CDC.
The theory going around, that has been tested and hasn't proven anything yet, is that newer vaccine ingredients contain a chemical that may be causing some kind of subtle brain damage. Not just post-birth vaccinations but stuff like the Flu vaccines given to expectant-mothers.
Another theory (along the same lines, but different factors) is that increased cellular radiation from the rise of smartphones and other mobile devices, especially cell towers, is causing these disorders.
There are really two issues:
1. For vaccines, you have to juggle the seriousness of getting an deadly or disfiguring disease vs. possible Autism. Studies and tests have been done, but there is no link between the two from everything that's been done. Doesn't mean there isn't a link, but there's no definitive proof yet.
2. For cellular radiation, you'll have to find a way to get past Verizon, AT&T, Sprint & T-mobile's lawyers, not to mention Apple, Google and everyone else who loves smartphones and the like, to even suggest cellular radiation could be causing these things. Imagine saying, "Sorry, world. Cell towers are causing Autism. Bye bye smartphones." Consumers and companies would flat out deny and ignore it even if it WAS true.
It's one of those... "Nobody really knows 100% for sure, and nobody wants to find out for 100% for sure because it'll cost billions of dollars and jobs and possibly end an industry and create lots of fear and panic and anger" type of things.
So far as vaccines are concerned, there are so many deadly diseases and viruses out there and there's a DEFINITE HIGHER CHANCE that not getting vaccinated might wind up killing you if you get measles, rubella, polio, etc... etc... So the Autism risk, even if there actually IS a link, is worth it to get vaccinated.
Fear is a powerful weapon, for BOTH sides of the debate.