I built a similar system for my school’s CS club. I considered using a door sensor, but the eventual solution I settled on was a light sensor, because it’s almost always true for us that the lights are on iff the door is open.
This way, we don’t need to mount anything on the door, we just have a microcontroller plugged into one of the machines.
Our previous solution was a webcam that pointed to the lights that did a similar thing (implemented by someone before my time) but then it stopped working due to some driver issues, and I didn’t want to spend time investigating them.