What happens if the flight price changes between the time my customer searches and the time they actually pay?

Yes, the price your customer sees in the search results can change before the booking is paid for, and a flight booking API is built to tell you about that rather than let it surprise you afterwards.

Airline fares are live. The seats and fares that come back in a search are what was available at that moment, and they can be taken by someone else while your customer is still typing in passenger details. Because of this, a flight booking API has a price check step, sometimes called revalidation, that you call after the customer has picked a flight and before you confirm the booking. That step goes back to the airline or the supplier and returns the fare as it stands right then, together with the current rules for baggage and changes.

How a change is handled depends on the supplier and on the airline. Some will simply return the new fare and let you carry on, some will refuse the booking and ask you to search again, and some will hold the quoted fare for a short window once the booking has been started. Your portal can be set to show the new fare to the customer and ask them to accept it before payment is taken, which is the safest way to avoid an argument later. Whether all of that is available to you depends on the suppliers connected to your account and on your plan.

Before you go live, ask each of your suppliers what their price check step returns and how long a quoted fare stays valid, and make sure your booking screen asks the customer to confirm once more if the fare has moved.