Hi there!
 I went on a medical mission trip before I transferred to University of Michigan so I can’t speak to the exact timeline for UM organizations. However, my application was due on October 1 for a trip in January—this is for a trip to Nicaragua—so I have a feeling that applications will be due soon. The considerable time between the application and the trip is to give students time to fundraise, gather medical supplies, and most importantly, get all the necessary paperwork/vaccinations to go abroad. Obviously, if the trip is domestic, the process may be a little shorter because there is less paperwork. 
I went with Global Brigades on a combined medical, water, and public health brigade. This consisted of 3 days of work in a clinic, 2 days of helping install a water system in a village, and 2 days of building an outhouse/installing a sewer system for a family. Again, I went with a chapter from a different college, but I know Michigan has a Global Brigades chapter
I chose to go with Global Brigades because they focus on the overall development of a community. The organization offers separate business, water, public health, and medical brigades, or you could do a combination trip like I did. Having different types of brigades helps build all foundational institutions in a community evenly so that the effects of brigades are more permanent. For example, a village could be served by a medical brigade frequently, but it does not have clean water and is therefore ever reliant on the brigade’s medical services. The medical part of the brigade consisted of screening patients i.e. recording medical history, teaching public health lessons to the children (how to brush your teeth), and gathering prescriptions for the pharmacist. 
Another one of the premier mission orgs on campus is MESO Their trips consist of 5 days of clinical service working in a clinic doing similar activities to those I described above. Unfortunately, the deadline to go on their spring break trip this year has already passed, since you have to pay dues in December. This is something to keep in mind for next year though! I would reach out now to express your interest for next year and/or to get acquainted with the org leadership. Sometimes medical service orgs have teams that stay in the country but are the point people for supply donations, which is a great option while waiting for next spring break to come around. 
Another suggestion I have is to browse Maizepages for other service organizations, reach out, get an idea of their mission/goals, then choose which meetings you want to attend. 
It’s important to note with medical mission trips that it’s unethical and it looks unfavorably upon your medical school application if you perform invasive procedures without appropriate training. Usually the organizations will ask what certifications, if you have any, on your application. This is why gathering a fair amount of information on the org before you pay the mission dues is very important !
Good luck! Feel free to comment more questions as they arise. 
Anni (advisor)