Technically, bodybuilding is a business. For the NPC & other sanctioning amateur organizations, it's nonprofit. For the IFBB, it is a for-profit enterprise, mainly benefiting the IFBB itself.
The act of posing onstage in front of spectators for a ranking is competition, & is probably closest to performance art, although it is held to tight limits on how to move & self-present.
If you consider a sport to be what is reported on generally as a sport, then bodybuilding unquestionably is not one.