The military determines qualifications for officers who receive direct appointment because of membership in a profession.
They must be physically fit and capable, they must have a masters degree or equivalent, they must be able to pass a background check, etc.
The army decides on the basis of "best qualified." The best fit, the best educational background, the best background results....
They then permit any religious group to contact them and develop a relationship as an endorsing agency. The endorsing agency will endorse candidates to be chaplains in the military who meet the military's requirements above AND who are fully qualified to lead religious services for their particular brand of religion.
The military also knows the religions of the soldiers it has in the force. That information is taken as part of the processing of all soldiers. There is a concerted effort to take the best qualified candidates, but there is also a need to have available chaplains who are of the same faith group as the soldiers in the military.
For example, if there are 250,000 Catholics in the military and only 1 member of "The Church of the Distant Rising UFO," in the military, it is critical that the military have far more Catholic priests than CODR pastors...if they should have any.
To avoid questions of favoritism, the military has wisely chosen to focus on what its needs are and on who are the most able to endure the military demands.
I haven't availed myself of the history of how the military was Federalized from State militias or when (my guess it was after 1933), but it looks like it might be an interesting investigation.