Autism doesn't actually manifest until about eighteen months into a child's life because it takes that long for the brain subsystems involved to be wired up in more or less their final configuration.
Funnily enough, that's about the same time frame as when the kid receives their first vaccinations.
To the nonscientific mind, correlation=causation a lot of the time. Add to that the increased difficulty in caring for an autistic child, the belief that autistic people are "soulless" or somehow not capable of empathy, the desire of parents for their children to live a "normal" life, and the significant subset of parents who view their children as lifestyle accessories rather than independent people and so are embarrassed by a child with special needs, and you have enough fear and dread that things might've been different had I made the choice not to vaccinate to kick off a worldwide movement.