No, you do not have to be a vet, but many do have a vet that they work with on a regular basis. Most wildlife rehabilitators don't have a degree, most learn their skills by volunteering/interning with already established trained rehabilitators. Some do have some formal education but there really is no such thing as a degree in wildlife rehab. Look here for info on how to become one:
http://www.tc.umn.edu/~devo0028/
http://www.wikihow.com/Become-a-Wildlife-Rehabilitator