Can the price change between the time I search a flight and the time my customer pays for it?

Yes, the price can change between your search and the final payment, and this is normal in air travel rather than a fault in the system. Search results are built from fare and seat data that airlines share in bulk, so they show you what was true a short while ago rather than what is true at this exact second. If the cheapest seats on that flight sell out while your customer is thinking it over, or if the airline reprices the fare or the taxes, the amount you saw in the search will no longer be the amount the airline will accept.

To protect you from this, the flight booking API checks the fare again with the airline before the ticket is issued. If nothing has moved, the booking goes through at the same amount. If something has moved, the response tells you the current amount so you can decide whether to go ahead or go back to your customer. Your own markup rules are applied on top of whatever the airline confirms, so your margin is not lost when a fare shifts.

How often this happens depends on the airline and the supplier behind the fare. Low cost carriers usually price live, so changes there tend to be rarer but can still appear when the last seats in a bucket go. Fares from the wider distribution channels sit in shared caches for longer, so they move more often, especially on busy routes and in peak season.

Build the fare check into your booking flow and only take money once you have the confirmed amount in hand, and tell your staff that a quote stands as an estimate until the ticket is actually issued.