The usual story (I feel like this thread appears every year) is that a student has a rotation with a preceptor. They end up getting a bunch of interviews, and ask the preceptor if they can go to their interviews. "Sure!" they say, and off they go to their interviews. Then, the medical school admin finds out about it and fails the student if they fall below a minimum level of attendance. So you can see how a good student could get into trouble.
Schools do this to be fair about it. What happens if you and your buddy are both on two identical rotations with different preceptors. You both get inteviews and ask for days off. One preceptor says yes, the other says no. How is that fair? Plus, preceptors may actually be unhappy with students taking days off, but feel like they have to say "yes" to these requests. So the school sets a maximum to make the system fair. The problem is that there is a spectrum between what's "fair" and what's "right". Fair = rules based, everyone is treated the same, logical. Right = emotion based, individualized based upon circumstances.
Anyway, back to the OP's problem:
As stated above, your first plan is to FIX this. Do whatever it takes to fix it. Work weekends. Clean toilets. Whatever. Agreed that if your school has an ombudsperson, you should contact them ASAP. As mentioned above, maybe you can make it up and still finish on time.
Failing that, you need to clarify whether your school will be withdrawing you from the match. You need to verify when your graduation date will be. if they will not withdraw you from the match and you will simply graduate a bit late, then you need to contact all of your programs and tell them. Some might drop you -- that's life. Many might keep you in the match, and start you late. Theoretically your school should NOT do this, but who knows.
If your school will be withdrawing you from the match, then you are in big trouble. You will not qualify for the match or for SOAP. You can try to get a spot outside the match after all of that, but there will be little left. You could ask your home program to create a prelim position for you (if they are willing) and give it to you after the match. As mentioned above, you could delay graduation for a year (perhaps getting an MS or MPH -- sometimes there are awards to fund this). Or you can graduate, look for off cycle spots, plan to match next year, apply all over again, and try to get some sort of job in the meanwhile -- research most likely.