If a customer starts a booking on my website and leaves before paying, can they come back later and finish it?

Yes, in most cases a customer can come back and finish a booking they left halfway, but how much of it survives depends on where they stopped.

If they only got as far as searching, what comes back is the search itself, the dates, the city and the travellers. The portal can save that against their account, or send them a link that reopens the same search. The prices though are fetched fresh from the supplier when they return, so what they see the second time may not match what they saw the first time. That is normal for live travel inventory and is not something the portal can hold on its own.

If they got further and the booking was actually held with the supplier, then there is usually a time limit, and that limit is set by the supplier rather than by you. A hotel might hold a room for a few hours, a flight seat often for much less. Once that window passes the hold drops and the customer has to start again from a fresh search. Some suppliers do not allow a hold at all, and the booking only exists once it has been paid for.

The part you do control is the follow up. The unfinished booking can sit in your back office where your team can see it, and the portal can email the customer a link to pick up where they left off. Whether that goes out automatically or whether you would rather a salesperson call is a setting you choose. Check with your team which suppliers you sell most often and how long each one holds a booking, because that is what decides how quickly the reminder needs to reach the customer.