Can my customer cancel their own booking on my portal and get a refund?

Yes, your customer can cancel their own booking on your portal, but whether money comes back, and how much, is decided by the airline or hotel supplier and not by your website.

When a customer presses cancel, the portal passes the request through to the supplier that holds the booking and waits for their answer. The supplier checks the fare or the room rate that was actually booked and replies with what is refundable. A flexible hotel rate cancelled before its deadline usually comes back in full, while a cheap non refundable air ticket may return only the taxes, or nothing at all. Your portal shows the customer that answer rather than an amount you have decided yourself.

The money goes back the way it came in. If card payments land in your own account, the refund leaves your account once the supplier confirms the cancellation, so allow for the gap between that confirmation and the bank putting the amount back on your customer's card. That gap belongs to the card networks and the banks, and no travel system can shorten it. Where the supplier collects the payment instead, the refund comes from them on their own schedule.

You can also switch self service cancellation off and ask customers to call your office instead, which many agencies prefer for anything already ticketed. Decide which of the two suits the way you work, then read the cancellation terms of each supplier you are connected to, because those terms, and not the portal, set what your customer will actually receive.