No one ever claimed that the MMR is the sole and exclusive trigger of autism. It's not just about the MMR, it's about the aggressiveness of the entire schedule, and the way vaccinations are intended to work. Even vaccine opponents aren't sure why the MMR seems to have a higher risk than other vaccines.
It is, in short, all about the
adjuvants. The immune response is an extremely complex biological process, but in layman's terms, it's somewhat like this: the immune system works in stages. The first antibody response is supposed to be from IgA antibodies, which live almost exclusively in the lungs and digestive tract. This is how most diseases enter the body. The IgA antibodies are the ones that organize the overall response to the pathogen, and call in the IgE antibodies (which are what give you a typical allergic response of itchy eyes, sneezing, etc.) to do the dirty work of fighting the pathogen. They also are the ones that best "remember" the pathogen for the future, so that the body has immunity.
When you inject the pathogen directly into the bloodstream, you completely bypass this IgA part of the immune system (and that is one reason why blood-borne diseases are usually so much more devastating than airborne.)
This is an acknowledged drawback of injected vaccines, and it is precisely why they include an adjuvant with the injection, usually the heavy metal aluminum. An adjuvant is a substance which, for an unknown reason, hyperstimulates the IgE response. We can't get the IgAs to remember the pathogen, since we've excluded them from the process, so instead we just throw the IgEs into overdrive, significantly overproduce antibodies to fight the pathogen, and that way plenty are left over after the disease is defeated. But these antibodies die off over time, and this is why injections don't provide lifelong immunity while getting the actual disease does, and why we need booster shots after a few years.
Is it really so hard to imagine that artificially stimulating the IgE antibodies would lead them to be more likely to mistake harmless substances (like, say, peanut proteins) as pathogens, thus creating an allergy to peanuts? It's not just about the MMR, and it's not just about autism. All autoimmune disorders are sharply on the rise, because we are screwing with our immune system and forcing it to work in ways it isn't designed to. The only reason the autism advocates are the only ones screaming is because having a kid with a severe peanut allergy just isn't that big a deal (I know, I have a stepson with one.) All the lesser diseases, while they might be lifestyle-altering, still leave you with your kid intact. But they are all a reflection of the problem--asthma, allergies, autism, ADHD, celiac disease, type 1 diabetes, rheumatoid arthritis...
Incidentally, it's quite possible to administer vaccines in an airborne fashion, breathing them in through a mask. When they do this, no adjuvant is needed, and indeed, that's how they do vaccinations in Africa, because they have to re-use all their medical equipment down there, and re-using needles in an AIDS-ridden population is a big no-no. Guess who doesn't have an autism epidemic like we do? Of course, detractors will say that's because African doctors are too stupid to recognize autism when they see it, and African parents are too stupid to know whether their child knows how to talk or not.