Can I follow the progress of a refund after my customer cancels, or do I have to chase the supplier myself?

Yes, you can track it, up to a point. When a cancellation is made, the booking is marked as cancelled and the amount the airline or the supplier has agreed to return is recorded against it, so you and your customer both have something in writing from the moment it happens. Your reports will show it as a cancelled sale instead of leaving it sitting in your takings.

What the software cannot do is make the money arrive. The refund is paid out by the airline or the hotel supplier on their own timetable, and that is quick with some and slow with others, especially where a person at their end has to review it first. The API passes on whatever status that supplier publishes, and where a supplier publishes nothing, the honest answer is that someone in your office will still have to follow it up.

It also helps to keep in mind that a refund and a cancellation charge are two different things. What comes back is what is left after the supplier has taken its charge and after anything you keep as your own service fee, so the amount your customer is told to expect should be the net one. Agreeing that clearly at the start saves a lot of difficult conversations later.

A good next step is to make one test cancellation with each supplier you use and note how much status information you really get back, so you know in advance which ones your team will have to chase by hand.