My school didn't have a committee either. What I did was ask each professor to send their letter to the med school, and to make it easier on them, I got envelopes with the dept. letterhead from them, printed the addresses on the envelopes, and stamped them. That way, all my writers had to do was put in their letter. That said, I turned in 28 secondaries (or smthg of that sort) and it became quite a hassle to keep track of every professor's letters. Obviously, they didn't get there all at the same time, but none of them got lost (not to my knowledge at least).
The frustrating thing was calling a school and having them tell you they're still waiting for a letter from one of the profs, even though your other letters had been there for months already, ya know? but it is definitely OK for them to arrive at diff times, the schools keep them all in your file.
If you're applying to a lot of schools, I would recommend using a service, and by August, I wished I had known about them before I started. It was SOO frustrating having to tell all those slacker profs over and over again to send them. Using a service, you only ask them to send it once, and you're done. for me, it was like....oh this person's on vacation for 3 weeks, he'll send them when he comes back, and trust me, during the secondary craze, the last thing you want to stress about is whether your letters have been sent. it takes them long enough to write them anyway. UGh I am so glad to have all that done with.
but if you trust that your writers will send them every time, and that they won't have a problem with it, the med schools don't mind. good luck!