They have done a lot of research on this. You actually have a better immune response when the body is confronted with more. Example you give adults Tetanus with Diptheria because it augments the response to Tetanus. The only live vaccines that are 'live' are MMR and varicella (chickenpox). The rest you are giving the equivilent of an empty shell to show the body what it should respond to. You give the immunizations early because the immune response happens but to have sustained response you need to remind the body. Many parents delay or stagger vaccines but there is absolutely no evidence to show it makes any difference except for make the parents happy. We don't have a very good record system or compliance in the country as a whole (Red states suck at this, esp down south). In countries with robust healthcare systems with standardized record keeping they have huge cohorts to get data from and it just doesn't play out.
There is no evidence at all that shows triggers from vaccines, no matter if a person is predisposed. As someone who did pedi for close to 30 yrs the observation I would make is Autism becomes more easily evident at around 15-18 months when you would expect the baby to be more interactive (the same time MMR is given). If you ask closely you usually will find that the baby has been 'off' from very early on but the parents just thought it was their baby's personality. They don't like to be cuddled, they weren't easily soothed, they didn't make eye contact or would only do so briefly, etc. There is also a strong genetic component. This means you have families that think behavior that is on the Spectrum is normal until it is diagnosed- usually when they don't meet markers that look at social cues- imitation, engagement with surroundings, etc
in my extended family we have a kid dx with autism. When he was around 18 months the mother thought something was not right. Trip to the Pedi and they suggested hearing problems, ADHD (person who said this was an idiot). They had a party and the grandmother grabbed me as we walked into the house and asked me to look at the kid. Huge party. The kid was in the middle not paying attention to any of it. It took me 2 seconds to say oh-oh. Long story short- the father was also dx. The father's mother couldn't understand why he had a dx. When you looked at the family there were 3 generations of people with the same personality traits. To them it was completely normal. They all thought the kid with autism was 'fine'. So how many people do we have who think their kid is normal because they don't know what normal is and miss cues until they are glaringly evident?