Using 555 timers, I managed to get a button locker system for this project. The left switch resets(left) or allows answers(right).
It works like this:
The Host flips the switch to the right.
One person pushes a button.
Their OR gate goes low.
Their 555 timer output goes high.
The OR gate on the far left locks both OR gates on.
If the other person attempts to push their button, nothing happens.