This is a very complex question. First of all, no one would say a vaccine "causes" autism. If it did, then everyone who got the vaccine would develop autism.
The proper question is: Is there a causal connection between receiving a vaccine and the onset of autism? And would these people (usually children) not develop autism without the vaccine?
There are four parts to the answer to that question:
1) Anecdotal evidence, such as, "My child was fine. Then he got is multiple vaccine. He was diagnosed with autism a month later, is common.
2) General associative evidence, that the rate of autism has increased as more complex vaccines have become more common, also exists.
3) All specific scientific evidence in detailed, careful statistical studies finds that there is no real correlation. There only appears to be a correlation because the age of diagnosis of autism is also the age of vaccination.
4) The scientific evidence has been challenged, and the challenge is a reasonable one. Each study, in itself, appears accurate. But the studies may well have been defined too narrowly (perhaps on purpose) to find the real problem. The studies were done primarily, perhaps entirely, on individual vaccines, but most vaccines, like MMR, are multiple vaccines. These have not been adequately tested.
Also, research has not yet broken out results by the way the vaccine is prepared (for example, from human tissue), by batches or companies of production, and a host of other variables.
As a result, I believe that the general and specific safety of multiple vaccines such as MMR in relation to onset of autism spectrum disorders is still uncertain.