I was in the same boat as you re: LORs. My school doesn't offer electives during third year either, so this is what I did (I'm applying this cycle) to get my letters:
1.) I did my Sub-I in internal medicine as the first rotation of 4th year. Got a rec
2.) Did back-to-back 4 week PM&R away rotations from the end of July to Mid Sept. Got a rec from each. The first rotation did the weekly thing. I made an appointment with the PD (who I spent a week with) asking if she would write a letter for me by using a compilation of the remarks from the 3 other attendings I worked with. She was happy to do this for me. Just let them know ahead of time that you would like a LOR.
My best advice would be to make sure you do your PM&R rotations early so you can get your letters in on time!