If the flight price changes between my customer searching and paying, will the API warn me before the booking goes through?

Yes, in almost all cases you will be told before the ticket is issued. Airline fares are live and are only held for a short moment, so before a booking is confirmed the API runs a check back with the airline to confirm the fare is still available. If the fare has moved, that check comes back with the new amount rather than quietly issuing at a different price.

What happens after that depends on how your portal is set up. The usual approach is to show the customer the new amount and ask them to accept it before any payment is taken. Some agencies prefer to let very small differences go through automatically and only stop the booking when the change is larger, so that customers are not interrupted over a few pennies. Either way it is a setting you choose, not something fixed.

Be aware that airlines do not all behave the same. Low cost carriers and direct airline connections can move quickly, and sometimes the result is not a higher fare but the seat being gone altogether, in which case the customer has to pick another flight. If you have already collected money before this check runs, you may need to refund or request the difference, and how smoothly that works depends on your payment setup and your plan.

It is worth asking us which re-price behaviour your setup currently uses and what tolerance, if any, is applied, and making sure your booking page carries a short line telling customers the fare is confirmed with the airline at the final step.