Can my customer hold a flight and pay for it a day or two later on my portal?

Sometimes, and it depends entirely on the airline and the fare, so it is not something you can offer on every booking. Some airlines allow a booking to be held for a set period before the ticket has to be issued, and when that option comes back through the API your portal can create the booking, show the traveller the deadline and wait for payment. Many fares, especially the cheapest ones and most budget airline fares, must be paid and ticketed straight away, and for those no hold is possible.

When a hold is allowed, the airline sets the deadline, not you and not the portal. If the payment does not arrive before that moment, the airline cancels the held booking automatically and the seat goes back on sale. The fare can also be gone by then, so a traveller who comes back late may have to be quoted again at whatever price is available at that time.

Some agencies prefer to handle this themselves rather than rely on the airline. They take a deposit or a booking form from the traveller, then issue the ticket right away so the fare is locked, and collect the balance afterwards. That is a commercial decision for your business and it carries the risk that the customer does not pay, so it suits regular clients and corporate accounts better than walk in enquiries.

Check with each of your airline and consolidator sources which of their fares allow a hold and how long it lasts, then decide whether you want your staff to offer it or to quote and ticket in one go.