Once my customer pays, can the ticket be issued automatically instead of my staff doing it by hand?

Yes, in most cases the ticket can go out on its own as soon as the payment is confirmed, with nobody on your team touching it. The system waits for the payment gateway to say the money has gone through, then sends the ticketing request to the supplier and saves the ticket number against the booking. Your customer receives the ticket by email or on WhatsApp, and your staff simply see the booking in the day's list as already done.

Whether this works every time depends on the supplier the booking sits with. Airlines you are connected to directly usually issue instantly. Some consolidators and some agency bookings still put the request in a queue that a person on their side has to clear, so those can take a few minutes or wait until their office opens. Low cost airlines normally confirm at the moment of booking, so there is nothing separate to issue. Your own balance with the supplier also has to be funded, because a ticketing attempt against an empty account will fail.

It is worth holding back the cases that should not go out unattended. Group bookings, fares that need a document check, and anything where the fare came back different at the last step are better seen by a person first, and the system can park those and tell your team instead. You should also decide what happens when issuing fails, so that booking lands in a list somebody watches rather than sitting quietly.

Ask which of your suppliers support ticketing through the connection you use, switch automatic issue on for those first, and watch a few bookings run through before you leave it to itself.