This is definitely something you will need to investigate and find a reliable answer to before beginning school (more reliable than we can give you on here). To work in Ontario, I had to prove through blood tests that I had received the MMR vaccine and it worked (I suspect that this is dictated my the Ministry of Health, but I don't have the time at the moment to try to look this up).
It turned out for me that my immunity was actually not that great to one of them so I had to go back and be vaccinated again. So my concern for you would not be about a specific allergic reaction but rather that you have immunity to these things. I don't know whether there is any flexibility in this or what your other options might be in terms of immunization.
You also mentioned paralysis. If there are lingering effects from this that will affect your ability to lift, then that will be an even bigger deal than everything else. Sometimes we don't think too much about that part of the job when entering a paramedic program, but it is one thing we do on nearly every call. If you can't lift, you can't be a paramedic.
As mobey says though, get in touch with your program coordinator (Mary Osinga, 705-749-5530 ext. 1733 or firstname.lastname@example.org). Don't trust anything that any of us have told you here. Even when getting information from Mary, be absolutely sure she is giving you genuine answers that can be backed up with documentation and if not then seek answers elsewhere. Something that could prevent you from having a career after two hard years of school is important to figure out now. I'm glad that you're trying to do this!
Good luck! And congratulations on getting into a paramedic program.