Can the flight booking API give me an e ticket I can email straight to my customer?

Yes, once a ticket has actually been issued the flight booking API sends back the ticket details, and those can be turned into a document you email to your customer.

What comes back is usually the ticket number for each passenger, the airline's own booking reference, the flight times, the fare and the baggage that goes with it. Some suppliers hand over a finished document ready to forward. Others hand over the details only, and your own system builds the e ticket from them, which is the better option if you want your agency name and logo on it rather than the supplier's.

There are two limits worth knowing. A ticket number only exists after the booking has been ticketed, so if you have only held or confirmed the seat there is nothing yet to send, and the customer should be given the booking reference in the meantime. Budget airlines often do not issue a ticket number at all, so what you send is a confirmation with the airline reference on it, which is what the airport will ask for anyway. How quickly the ticket details appear also depends on the supplier and the airline.

Make a test booking on a route you sell often, look at exactly what the supplier returns, and check that your e ticket template picks up every field before you rely on it with a real customer.