If I have to cancel a flight ticket I already issued, can I do the refund through the software or do I have to deal with the airline myself?

Yes, in most cases you can start the cancellation and the refund request from inside the software, and the request goes to the airline through the API without you having to call or email them. You open the booking, choose the cancel or refund option, and the system sends it on and records what came back. What you cannot do is decide the outcome, because the airline rules decide how much comes back and when.

It helps to know there are really two different things here. If you act very soon after issuing, within the window the airline allows for the same day, the ticket can often be voided, which means it is treated as though it was never issued and very little or nothing is held back. After that window closes it becomes a proper refund, and the fare rules take over. The airline may keep a cancellation charge, may keep some taxes, and in the case of a fully non refundable fare may return nothing at all beyond the odd recoverable tax.

Be aware that not every airline supports this automatically. Low cost carriers and some fare types do not offer an automated refund through the API, so the request becomes a manual one that is handled by the supplier or the airline directly, which takes longer. The refund figure you see on screen can also be an estimate until the airline confirms it, and the money normally goes back the same way the booking was paid, so your customer will not get it instantly.

Before you issue anything, read the fare rules on the booking so you know what is refundable, and once a cancellation is in, tell your customer the amount is confirmed only when the airline confirms it.