After consulting m-w about 'accoutrement', it seems that word is about ancillary equipment. However, it suggests there might be an appropriate short phrase we might use to describe what you're saying:
Essential equipment or Essential element might be close, but these would connote both, for example, a chess board and the pieces whereas I think you're looking for a phrase that would describe the pieces, but not the board (the soccer ball, but not the field).

Manipulated entity characteristic of the game. (Yuck)

Fundamental item of play.


Heck. I don't know.