The idea of inspection of the vault is that something happens in the vault (riot/breakout) and you have to come for an inspection to fix all the weak spot it may have. Initially the inspection of the vault had crisis icon but i realized that it's much more fun this way because you can get a false feeling, that you don't need to do the inspection but then you realize that if you don't and vault cards are revealed from the encounter deck, you get seriously punished. If the inspection of vault had crisis or hazard, you d immediately (if possible) resolve the obligation and very rarely the "if inspection of the vault is in play" would happen.
It was a cool game. Unfortunately there seem to be a lot of inaccuracies in the insert and on the cards. I was wondering why your co-player kept the obligation for that long... You are playing with the fire 😂😂😂