The age when symptoms of autism set might be consistent, or it could be just the age at which they are most easily recognised (to an extent that's literally what a symptom is anyway though). The young age at which most people get diagnosed is largely due to the imbalance of support given to young children – in a lot of countries the probability of a child being accurately diagnosed plummets as soon as they hit 13, and almost nobody is diagnosed in the second half of their life.

No number of vaccines is ever going to change anyone's DNA or give/remove a geneic condition like autism. If there was then humanity would have already cured disorders like Down's syndrome which have, in comparison, extremely clear chromosomal indicators.

There's an argument that vaccines could "exaggerate symptoms" of autism, though of course almost none of the thousands of studies on links between autism symptoms and vaccine administration have ever picked it up because it would be impossible to test when so many other factors do the same, or the effects would be so tiny that finding a conclusive result wouldn't even be useful in the real world.