Can I let my customer pay part of the money when they book and the rest closer to the travel date?

Yes, on some bookings, but not on all of them, and the supplier decides which ones. A part payment works where the supplier does not need the whole amount straight away, which is common with hotels that allow payment near the stay date and with some tour and holiday products. Flights are usually the opposite, because most airlines want the fare paid and the ticket issued at the moment of booking, so there is nothing left to collect later.

On the portal you can set this up as a deposit rule. Your customer pays the agreed part when they book, the balance is recorded against the booking as still owing, and you choose the date by which it has to be paid. The software keeps that balance visible to your staff and can prompt you and the customer as the date gets close, so nothing is forgotten in the weeks between booking and travel.

Be clear with the customer about what a part payment really buys. Until the supplier has the full amount, the booking often sits as held or on request rather than fully confirmed, and if the balance is not paid in time the supplier can release it. Whether a balance due later is allowed at all differs from one supplier and one API to the next, and card charges and refunds on a part payment follow the terms of your own payment provider.

The sensible next step is to ask each of your suppliers which of their products allow a balance to be paid later, then set your deposit rules and reminder dates on the portal to match what they actually permit, and write your payment terms onto the booking confirmation so the customer knows when the rest is due.