Why not vaccinate children at 5 years old?

I don't believe vaccines cause autism and I believe vaccines are a net-positive for humanity. But, could studies be better done if we have a substantial population that is vaccinated later? Could we remove the effect of vaccines on children to determine if other more correlated factors cause autism?

Autism is correlated to vaccination rates. But I am wondering if it is due to the timing. The present vaccinations were done were done later in life in terms of age. Maybe delaying of vaccines is better?