Peer-reviewed, published research does not support any kind of link between vaccines and autism: not a link by correlation, not a link by cause.
There was a paper a long time ago where the guy who wrote it committed fraud to get published. This paper, which was founded on lies, did claim a link, and did so in a highly influential medical journal - so a lot of people paid attention to what the article said, even though its claims were never true. Retraction is a very definitive way to signal to scientists not to pay attention to a particular paper.
Problem is, the retraction took more than 10 years, and in those years the internet was becoming social and available to everyone. So there was kind of a perfect storm of misinformation propagation about this particular topic.
Fact remains: many many studies have re-examined the possibility of link between vaccines and autism and there just isn't one.