They will often write it, althoug it may be a stock letter. You could always try to chat them up once or twice in the next few days or something like that, so at least it's not coming out of NOWHERE.
I had a friend (Johns Hopkins) whose committee required 7 LORs. If you're in that kind of situation, don't worry about it. I think when they ask for that many, they know they can't possibly all be people who know you really well. He ended up asking a teacher he had never met for one, and everything worked out great (got into top 10 school).
You gotta do what you gotta do.
The bright side is that you figured out a chink in your med school application before you even applied to med school (this is for an internship, right?). Great. Now go flirt with your professors for the rest of your undergrad career.