Can my customer book a flight that is leaving in a few hours, or is there a cut off?

Yes in most cases, though every airline sets its own cut off and once that point passes the flight simply stops appearing in the search results.

Airlines usually close sales somewhere between one and a few hours before departure, and the exact moment differs by airline, by route and sometimes by the airport. Your portal does not decide this. It shows what the supplier returns at that moment, so a flight that was there at breakfast may quietly drop off the list by lunchtime. For international departures the cut off tends to be earlier than for domestic ones, because check in and boarding are longer.

There is a second thing to watch on very late bookings, which is the time it takes for the ticket to actually be issued and reach the airline. Payment being taken is not the same as the passenger being ticketed, and on a rare occasion a booking made very close to departure can be accepted and then fail at the ticketing stage. That is why it is worth checking the booking on your portal rather than assuming, and telling the passenger to reach the airport early rather than just in time.

If same day selling is a regular part of your business, ask which of your flight suppliers handle last minute issuing best, and make a habit of confirming that the airline reference has come through before you tell your customer to set off.