NY-Catcher,
While, I wasn't in the Army (I'm retired from the Air Force), I spent over 10 years working in Joint assignments. I support an Army organization in my current job.
Based on my experiences and some checking here's what I've come up with:
The Army doesn't have an Army-wide team for baseball, but they do have one for softball. Check out
Army Sports for a listing of the sports.
However, several Army units have baseball teams that either compete on a post-level or are entered into local community adult leagues.
Several joint organizations (U.S. Southern Command in Miami, FL for one) have teams that are made of players from their organizations (Army, Navy, Air Force and Marines.) When I was stationed in Miami, one of my co-workers, who was in the Army played for the SOUTHCOM team as well as playing for a local adult team.
The Navy also has baseball teams, many are affiliated with either their particular base or ship. But they also have a Navy-wide baseball program, where sailors can try-out for the team. They are based out of San Diego and play in the Southern Division of the Western Baseball Association.
In 2005, a combined military baseball team representing all the armed forces and the U.S. Coast Guard competed aainst the Cape Cod and New England Collegiate Baseball Leagues.