Yes, in most cases you will get a confirmation number you can pass to your customer, though it is not always the hotel's own number straight away. A hotel booking made through an API normally comes back with two references. One is the booking reference created by the booking system, which is what you use to look the booking up, amend it or cancel it. The other is the supplier or hotel confirmation number, which is the one the property staff can usually find in their own system when your guest arrives.
For instantly confirmed bookings the hotel number is usually returned in the same response or shortly after, so you can put it on the voucher you send out. For bookings that go through on request, the room is held while the property replies, and the hotel number may only appear once that reply comes back. Until then your customer has a valid booking reference but not a number the hotel can quote, so it helps to explain that in your first confirmation message.
How much comes back does depend on the supplier behind the rate. Some return the property confirmation number, the guest name as the hotel holds it and the full address, while others return only their own reference and expect you to contact them if the hotel queries anything. If a supplier does not send a hotel number, we cannot create one for you.
Make a test booking on a rate you sell often, check which reference fields come back and how quickly, then build your voucher around the ones that are always filled in.