There were two connections with vaccinations and autism... one was a preservative that contained mercury that was being used in many vaccines. A few years ago, the government required use of that preservative be stopped.
The second connection was/is related to the MMR vaccine. Vaccinations either have active or inactive cells in them. MMR has live, active cells. Your baby doesn't get his/her first MMR shot until the 12 month visit. Some people are asking their peds to spread out the vaccines -- Measles, Mumps, Rubella -- giving them in 3 different visits rather than in one super shot. The argument is that our little babies don't have mature enough immune systems to fight 3 active cultures for 3 different diseases at one time. I think this is prudent and plan to do the same thing with my baby.
It is against the law to refuse the vaccination completely unless you can cite religion as the reason for refusal and you must file for the exception. You can delay vaccination but I'm not sure for how long...probably until your child enters school or daycare.
Don't be surprised if your pediatrician pushes you into doing the MMR supershot. Most medical doctors are trained to spout of statistics on vaccine safety.... and to the majority of children, vaccines are safe. But what if your child is that one out of one thousand.... that is what you want to prepare for and try to prevent.
One final note, do not take your child for a vaccine if he/she is coming down with a cold or getting over one-- make sure they are healthy! And don't be afraid to give them vitamin A&C supplements before and after the vaccination.