Yes, in most cases the system can work the refund out for you. When a booking is cancelled it looks at the cancellation rules held against that booking, applies the supplier penalty for the date the cancellation is made, keeps back your own service fee if you charge one, and shows the net amount due to the customer before anything is sent. Your staff see the figure and the reason for it on the same screen, so nobody has to work it out on paper.
How accurate that figure is depends on the supplier. Where the booking came through an API that returns live cancellation terms, the penalty is taken straight from the supplier and is reliable. Where the terms were only shown as text at the time of booking, or the booking was typed in by hand from a phone or walk in deal, someone on your team has to confirm the penalty before the refund is approved. The system will still do the arithmetic, but it can only be as right as the rule it has been given.
Paying the money back is a separate step from working it out. A refund can normally go to the card the customer paid with, or be credited to an agent balance if one of your agents made the booking, and you decide which of those you want. How long it takes to reach them is set by the payment gateway and the bank rather than by the system, and you may be refunding the customer before the supplier has refunded you, so that gap is worth watching.
Check which of your suppliers return cancellation terms automatically and which do not, and decide whether refunds should go out straight away or wait for your approval first.