Please vaccinate your baby.
Many rigorous, exhaustive studies have been done looking at a huge number of children, and there is no correlation between the MMR vaccine and autism. There is a lot of research going on into what causes autism, and a number of intriguing theories have been proposed, but nothing has been proven yet EXCEPT a non-correllation between vaccines and autism.
In fact, according to the Lancet, the doctor who first proposed such a link no longer practices medicine. His research has been retracted, and he's been disciplined for improper conduct. His "research" was hopelessly flawed and has fueled a debate over vaccination which really has no basis in science.
Vaccines do not "overload" the immune system, and there is no advantage to waiting, spreading the vaccines over a long period of time, or separating the vaccines. Honest to God. In fact, the vaccination schedule is designed to protect kids when they're most vulnerable and most likely to experience significant complications should they contract an infectious disease.
Measles KILLS. And it can cause lifelong disability. 30% of victims in the US this year will experience some complications of the disease, and 1-2 per 1000 will die. (Measles outbreaks pop up whereever clusters of parents are refusing vaccination, and we aren't always talking about Amish parents, or poor, uneducated families living in bad nutritional or hygienic situations. We're talking about wealthy people in places like Boulder, CO.) Rubella can cause all kinds of birth defects or even death in unborn children whose mothers catch the disease. There was just a multi-state outbreak of mumps THIS YEAR. Mumps can cause infertility, along with a host of other complications, including miscarriage and deafness. And it's all preventable! There's no reason it has to happen.
And the risk of a real reaction to a vaccine is so small that it's hard to actually quantify.
I didn't have the MMR separated, and I vaccinated my kids.
I hope you do, too.
Even if you decide that you're going to insist on separating the vaccines, or delaying their administration, vaccinating late is better than not vaccinating at all.