If my customer cancels a flight, can I do the cancellation and the refund myself on my portal?

Yes, in most cases you can cancel a flight ticket from your own portal and start the refund without emailing anyone, but how much comes back, and how quickly, depends on the airline and on the fare you sold.

When you cancel a booking, the request goes to the airline through the same connection that issued the ticket. The airline decides the refund based on its own rules for that fare, so a cheap promotional fare may return very little or nothing, while a flexible fare may return most of what was paid. Your booking screen shows you the cancellation charge before you confirm, so you can tell your customer what to expect instead of guessing.

Some cancellations cannot be done automatically. Low cost airlines often handle refunds on their own systems and on their own timelines, and tickets that are partly flown, already checked in, or past the departure time usually need a person to look at them. In those cases your portal still records the request and it is followed up with the airline, so the cancellation happens, just not in the same minute.

The money itself moves in two steps. The airline releases the refund first, and only then can it be passed back to your customer's card or to your credit balance, which is why a refund can show as approved on your side well before it reaches a bank account. Before you sell, read the fare rules shown with the search result and tell your customer plainly what is refundable, and always check the cancellation charge on the booking screen before you confirm.