Honestly, virtually no requirements as far as coaching. In some school districts, you are required to teach at the school or at least within the district. In other regions, typically more rural areas, one might not need anything more than the willingness to coach-provided they pass a background check, of course-in some areas this might not even happen.
Many schools have difficulty finding someone to coach some sports and new teachers, in some instances, are required to coach, for example, the first 2-3 years of their contract. From what I’ve heard, such a practice is actually becoming more infrequent.