I don’t know the mechanism on the actual cabinet but you could use a microswitch like [this](Cylewet 6Pcs V-153-1C25 Micro Limit Switch Long Straight Hinge Lever Arm SPDT Snap Action LOT for Arduino (Pack of 6) CYT1068 https://www.amazon.com/dp/B071NSRHK3/ref=cm_sw_r_cp_api_glt_i_TWGH51ZHCQ7MKZSNNH0T) inside of the holster. It would be in the pressed position when the gun is inside and in the off position when the gun is removed. You would just have to wire it correctly for the game.

Link isn’t working but search micro leaf switch on Amazon