The cook says we are out of the fish. Can they take it off themselves?

Yes, and they should. Press Something has run out on the kitchen screen, pick the dish, then either say how many are still possible or None left. It comes off your ordering page and your POS at once, and it comes back on its own at the end of your day, so nobody has to remember. Each one is written down, so you can see later what ran out and how often.