I believe the immunization rules don't apply to Americans. We're all imunized at birth for MMR and Rubella (german measles).
Also, haven't we erradicated polio?
There's nothing about immunization on the Britainusa website. I think this must be for countries that still have problems with these diseases. I'd check with the university though, it might be them that requier proof of imunization?)
Yes, you do apply online. It's easy, just have all of your info in front of you. Then you can pay online, print a copy. They'll tell you what documents to send and where to send them after that. (I'd send the yellow card with the application, since the visa is what says how long you can stay, not the IO. You'll get it back so you can bring it as well.)
Good Luck!!!!!