Once a customer has paid, does one of my staff still have to send out the ticket and the hotel voucher by hand?

No, in most cases that goes out on its own. When the payment clears and the supplier confirms the booking, the system puts the ticket or the voucher together with your own logo and contact details on it and emails it to the traveller, and keeps a copy against the booking so anyone in your office can send it again later. Your staff only get involved when something is unusual.

There are limits, and they nearly always come from the supplier rather than from your side. Some bookings confirm straight away and the document exists within seconds. Others come back as on request, and there is nothing to send until the supplier says yes, which can be the same day or the next morning. A few suppliers send their own voucher in their own format, and in that case what reaches the traveller is their document, not one carrying your branding.

You also keep control of when it goes. If you would rather nothing leaves the office until an agent has looked at it, that can be set so the document waits for a click instead of going automatically, which some agencies prefer for their higher value bookings while letting everything else run on its own.

Before you rely on this, send a test booking through each supplier you use and look at what actually lands in the traveller's inbox, because the answer is not the same for every one of them.