When my customer pays for a flight, does the ticket get issued straight away or do I have to do something myself?

Usually yes, the ticket is issued within seconds of the payment going through, but it does depend on how your account with the supplier has been set up.

On a normal instant ticketing setup, once the payment is confirmed the booking request goes to the airline, the seat is confirmed and a ticket number comes back in a few seconds. Your customer can be sent the electronic ticket right away and there is nothing more for you to do. This is what most agencies prefer, because nobody is left waiting and no seat is sitting unpaid.

Some accounts are deliberately set up the other way, where the booking is held first and ticketing is a separate step. That is common when you want to check the fare before committing, when the fare is on a deposit or a group contract, or when someone in your own team has to approve the payment first. The booking then shows as confirmed but not ticketed until a person issues it, and the airline gives a deadline after which the seat is released. There are also times when the airline itself is slow to answer or the fare needs a manual check, so a booking you expected to be instant takes longer than usual.

Whichever way you are set up, treat a booking as finished only when a ticket number has come back, not when the money has been taken. Ask whoever set up your account which of the two applies to your flights, and if it is the second one, make sure somebody is watching for bookings that are still waiting to be issued.