It has been shown that the evidence for the causal link between MMR vaccine and autism was weak at best, and incorrect at worst. Andrew Wakefield, who published this evidence has been exposed as fabricating his data. For a really quick overview and lots of extra reading on this go to his Wikipedia page.

Parental doubt and confusion is caused by the fact that first symptoms (if not diagnosis) are often observed at the time of, or shortly after, the vaccination, so there is a temporal link but not a causal link between vaccination and observation of first symptoms. Current research indicates some brain differences as early as six months in children who are not yet showing signs of autism but go on to do so in the second year of life.

– Dr Elfriede Ihsen