Can my staff put the bookings they take over the phone into the same system as my website bookings?

Yes. Most bookings in a travel business still start with a phone call or a message, and the back office is built with that in mind. Your staff can create a booking by hand, entering the customer, the supplier, the travel dates, what you are paying and what you are charging, and attach the supplier confirmation to it. Once it is saved it behaves like any other booking, so it appears in your sales reports, in that customer's history, in what you are owed and in your supplier settlement, instead of sitting in a separate spreadsheet.

The difference is that a manual booking cannot check itself. A booking made through your website was priced by the supplier, so the cost is whatever the supplier returned. A booking typed in by a person is only as accurate as the person typing it, so wrong cost prices and forgotten markup are the usual trouble. It helps to make the important fields compulsory, so a booking cannot be saved without a supplier, a cost and a reference number, and to have one person look over the day's manual entries before invoices go out.

There are limits worth knowing before you rely on it. A booking entered by hand normally has no live connection back to the supplier, so if the hotel later cancels or moves your customer to a different room, nobody is told automatically and your staff must update it themselves. Some suppliers will let their offline bookings be imported in bulk, but that depends entirely on whether they provide any usable file or feed, and on what your plan includes.

Ask your offline suppliers what form they can send you a booking list in, and agree with your team which fields must never be left blank before they start entering bookings by hand.