Due to your concerns, I would request that each vaccine in the MMR be given separately rather than all in one shot. That dramatically reduces the risks of vaccine injury.
EDIT: Huh, looks like you're right. They no longer spread them out. My 3 year old nephew had his separated because my SIL is so paranoid.
And now I see that there's an MMRV vaccine... Measles, Mumps, Rubella, and Varicella all in one. One reason that they should be together is that there are actually fewer chemicals that are harmful to the children. Fewer times the child has to be poked means less "filler" chemicals and additives and other ingredients in addition to the medication.
The problem with "links" is that when most people start showing signs of ASD and Sensory Processing Disorder and other related issues is the same time that the vaccinations are recommended to be given. The symptoms would be showing up regardless of those vaccinations. They're not causal, they're coincidental unless you can prove an allergy to an ingredient that was used. And mercury/thimiserol is no longer used. The other ingredients affect people in such small stats that they shouldn't be a concern. More children get sick and die from peanut allergies.
EDIT AGAIN: Andrew Wakefield was a quack. He was not debunked by a reporter, he was debunked by SEVERAL independent scientific studies. He wasn't just "given a bad rap." He was scientifically disproved over and over and over. He has lost his license to practice and is no longer a doctor. The man is scum. It goes far, far beyond a reporter breaking the news about what he did... faking results and faking subjects in his non-existent studies. He tried to connect MMR to causing Crohn's disease prior to Autism but his peers laughed at him and were all, "LOLWTF? Srsly? Nuh uh!" The Lancet where he originally published his hoax retracted the original article recently. This is real and this is serious. There's so much that the general public doesn't realize or know about this hoax that Wakefield perpetuated. But trust me, he was not simply "misunderstood." He was not persecuted unfairly. He was scientifically disproven and then brought to court and a judge took away his license to practice. Anyone who values his opinion after that... well... I feel pity.
He set back research on Autism for DECADES and my daughter has to be the one to deal with that. My friends' children have to deal with that.
Not only are autistic children suffering but parents all over the world including the United States have dead and/or severely ill children because they didn't get necessary vaccines for their children because of that pig.