How can you register for college classes without MMR immunization records?

Ask your college. Some colleges will allow you to register without proof of the immunization and some won't. If you can't find the actual shot record, they may ask that you have a blood test called a titer to determine if you are immune to either measels or rubella. Ask your college what they require that you have for attendance there.
Don't think possible. Measles is a highly contagious and potentially serious infection, and public and private schools as well as colleges require proof of immunization to protect students and staff. Mumps and rubella are also in the mmr. You'll need to be vaccinated.
Blood tests (titers) Blood tests can show whether or not you have immunity to mumps, measles (rubeola) and rubella (german measles), if you do not have the records to show immunization.
Religious exemption. Check with the college and see what is required. Some need only a waiver. Other than that get a religious exemption. This assumes you did not have mmr. If you did and cannot find your records, then you can have a blood test done checking for titers and that is proof of immunity.
Depends on state. If you were immunized blood tests may prove it. If you choose not to be immunized, most states allow exemptions to school-required vaccines based on religious and/or philosophical beliefs, but the laws vary from state to state. See http://www.Ncsl.Org/issues-research/health/school-immunization-exemption-state-laws.Aspx and http://www.Nvic.Org/vaccine-laws/state-vaccine-requirements.Aspx.
Get vaccinated! Mumps in post-pubertal men can cause orchitis (exquisitely-painful swelling of the testicles), which can lead to sterility. Measles can cause potentially-fatal encephalitis. Rubella, when passed on to a pregnant lady, can cause devastating birth defects. These are not diseases to be trifled with. Get vaccinated!
Immunization records. If you have chosen not to get immunized and that's why you don't have records, then ask the college if they have any type of waiver. If it's just you can't get your hands on records, ask your high school or even your elementary school if they have immunization dates if your physician cannot supply them. Also, many states now have immunization registries and you may be able to find your dates there.