I don't vaccinate my son, and it has nothing to do with autism!! I don't really believe autism is caused by vaccines. (Although, I do think chemicals in our body don't help children who are already autistic from what I've read. Not that I'm saying that's definite, or fact. I've just seen reports here and there. I don't think vaccines are a cause. There has been nothing linking vaccines to autism. Nothing I believe, anyway.) You can get the MMR vaccine separate, and that is actually recommended by many pediatricians. My pediatrician offers the MMR separate only. Many doctors won't allow it also, but it has more to do with convenience. You can just order one separate vaccine. So, if you and only you want it, they won't order it. If many wanted it, then they could order a small order of 10 or so. (I don't think 10 people at the same time will want that, though!)You CAN get the Dr. to write a prescription for the separate vaccines,have it filled through the pharmacy, and bring the unopened vaccine to the office. You have to ask a Dr. to do that, though. Many, many parents get all the vaccines on schedule, but wait for the MMR, so they can get it done at three different times, rather then one. It's not uncommon.
About the eggs. You'll have to ask your pediatrician if it causes an allergic reaction. The MMR vaccine does incubate in DNA from chicken cells that are collected from eggs. I don't know what that means for an egg allergy. That's really something your Dr. would have to consider.