This is the first time this ever occurred, I've always had my XB1 controller plugged into the PC but I don't use it to play Insurgency. What follows is my steps to repro the issue:
XB1 controller plugged in > Boot game > All menu controls XB buttons > Joined friend's party > Friend joins Co-op lobby > Get to class selection > Select with mouse > Select loadout, but can't "Deploy" > Hit X on controller to deploy > Can't move player or access ESC menu and had to ALT+F4 and restart.
Restarting the game with the controller unplugged fixed issue.