Will I get a confirmation number from the hotel itself when I book through a hotel booking API?

Yes in most cases, but it does not always arrive at the moment you make the booking. When a hotel booking API confirms a room it normally sends back a supplier reference straight away, and alongside it the hotel's own confirmation number when the property has given one.

The supplier reference is the one you can count on, because it is created by the system you booked through. The hotel confirmation number comes from the property itself, so whether you get it, and how quickly, depends on the supplier and on the hotel. Some properties pass it back within a minute, some take a few hours, and some never send one through the API at all. That does not mean the room is not held. It usually means the property confirms with the supplier in its own time rather than instantly.

For your customer this matters less than it sounds. Ask them to carry the voucher, because the voucher shows the reference the front desk will actually search for, and if a hotel number does arrive later it will normally appear on the booking in your system so you can send a fresh voucher. If a customer is nervous about it, most suppliers will check with the property directly if you ask them.

The thing to do next is make one test booking with your supplier and look at what comes back, so you know which references you receive, where they appear on the voucher, and which of your suppliers hotels tend to return a property number.