Yes, they certainly can. bookitlive integrates with a range of payment gateways, so just create an account with one of those services and add the details into your bookitlive account. Your clients can start payment for their bookings online right away.
There are a number of options available, too – make online payments optional or mandatory, ask for the full amount, or just take a deposit.
The payment processing gateways are integrated with bookitlive: