Yes, a refund can usually be sent back to the same card your customer paid with, but it is not one single automatic step and it helps to understand why.
A cancellation has two sides. The supplier or airline has to decide what it will give back, and that depends on the fare rules, the hotel's cancellation policy and how close you are to departure. Some suppliers confirm the refundable amount in seconds through the API, others take days and handle it by email. Your booking system can hold the cancellation and show it as pending, but it cannot hurry along a supplier who works by hand.
The other side is getting the money to your customer. If they paid by card through a payment gateway, the refund normally goes back to that same card and your staff trigger it from the booking once the supplier amount is known, so nobody has to ask for card details again. If they paid by bank transfer or in cash, the system records the refund as due but someone in your office still has to send the payment. Any cancellation charge you keep for yourself can be deducted on the same record, so your accounts and the customer's statement agree.
Check with your payment provider how long a refund takes to reach a cardholder and whether their processing fee comes back to you, then write your own cancellation terms to match, so what you promise the customer is what you can actually recover.