There is the "All Academy" Wrestling tournament that was held at Navy in February this year which featured USMA, USNA;USAFA;USMMA;USCGA, VMI & Citadel- and in March the same schools competed in Track &Field in the "Military championships" held this year at VMI.
The Tri-Services Powerlifting Meet had entries from all 5 Academies when I attended in 2007 at Annapolis. I haven't been to one since, but I would think that each year would be similar.
They all compete in boxing through the National Collegiate Boxing Association which was founded in 1976. (The NCAA discontinued their support and recognition of boxing in 1960).