Hello! I knew the OT I shadowed for one summer, and she wrote me a recommendation. If you make a good impression, you shouldn't have problems getting letters of recommendation. I asked two other professors who have known me for most of my undergraduate career. But that doesn't necessarily have to be the case.
Ideally, I would ask references a month in advance out of courtesy and I altered the due dates for some who I knew were forgetful so that if something happened and they didn't get it in by my fake deadline, all would not be lost. However, my professors were more than willing to submit last minute recommendations for other students. That happens to them pretty frequently, but I would avoid that if possible. The more warning, the better. I also supplied them with my resume, the classes I had them with, my personal statement, and why I wanted to get into each program.