Can my customer end up paying a different price from the one they saw when they searched?

Yes, it can happen, although on most bookings it does not. The prices on your search page are the ones the supplier gave at the moment that search ran. Airlines and hotel suppliers change their availability and their rates through the day, so by the time your customer has picked a room, filled in the passenger names and reached the payment page, that exact rate may no longer be on sale.

This is why a booking system checks the price with the supplier again just before taking payment. If the supplier comes back with the same price, the booking goes through quietly and your customer never knows anything happened. If it comes back different, a sensible system stops there and shows the new price for the customer to accept, rather than charging the card and leaving your team to sort out the difference afterwards.

How often this happens is up to the supplier and not up to you. Some suppliers hold a quoted rate for a short while after the search, some do not hold it at all, and budget airlines in particular change fares constantly. No booking system can make a supplier honour a price they have already withdrawn, so anyone who promises you the search price is always final is not being straight with you.

The thing to check with whoever runs your portal is that this recheck happens before the card is charged and not after, and that your team can see which bookings were repriced. If one supplier keeps doing it far more than the others, that is a conversation to have with them.