Can I hold a flight seat for a customer who says they will pay me tomorrow?

Sometimes yes, but it depends entirely on the airline and the fare you have chosen, not on us. Holding a booking without paying is an airline feature. Where an airline allows it, the booking is created and a ticketing time limit is attached to it, and the seat stays reserved until that deadline passes. Many full service airlines allow this on normal published fares. Most low cost airlines and most heavily discounted fares do not allow it at all, and those have to be paid and ticketed in the same session or not at all.

The important thing to understand is what a hold actually protects. It usually protects the seat, and in many cases the fare as well, but this is not guaranteed on every airline. Taxes and surcharges can still be recalculated at the moment you issue the ticket, so the amount you finally pay can differ a little from what you quoted. If the deadline passes without payment, the airline cancels the booking automatically and the seat goes back on sale, so your customer loses it quietly without anyone telling you.

How this appears to you also depends on your plan and on which airline content you are connected to, because some suppliers pass the ticketing deadline through and some do not. Where it is passed through, you will see the time limit on the booking so you know exactly how long you have.

Before you promise a customer anything, make the booking and check whether a ticketing time limit is shown on it, then set your own reminder well before that time so you can collect the money and issue the ticket in good time.