Can the fare change between the time my customer searches and the time they pay?

Yes, it can. Air fares are live, and the amount you see in a search result is only valid for as long as the airline or supplier keeps holding it. Between a search and a payment, seats in the cheapest fare class can sell out, the airline can reprice, or a currency rate can move. This is normal across the industry and not something any booking system can fully prevent.

What the flight booking API does is check the fare again before the ticket is issued. That repricing step returns the current amount, so your booking flow can compare it with what the customer was shown and act on the difference. Most agencies either show the customer the new amount and ask them to confirm, or quietly accept small differences and absorb them, and block anything larger. You decide which behaviour suits your business and build it into your checkout.

Whether you can hold a fare for a while depends entirely on the airline and the supplier. Some full service carriers allow a booking to be held for a limited period before ticketing, so the fare stays fixed in that window. Many low cost carriers price only at the moment of payment and offer no hold at all. We cannot promise a hold that the airline itself does not give, and what is available to you may also depend on which suppliers are enabled on your plan.

Decide your own rule for handling a price difference, then ask us to confirm which of your connected suppliers support a hold so you can set that rule sensibly.