If you go through a mission board, with a group from your community or your church, with a US physician dictating what you can do, you can go on short term trips that include medical clinics and basic medical care. There is need, but the situation is more complicated than "I want to do medical care."
By contacting an organization that supports missionaries (do you mean Christian missions work?), you can find orphanages, feeding centers and even hospitals in areas that need it who can use your help if you're looking for something more long-term.
Be forewarned that an EMT is going to become a jack-of-all trades in these situations, and not necessarily be doing medical care all of the time.