Can two of my staff work on the same booking at the same time without overwriting each other?

Usually yes, but it depends on what each person is doing. Most travel agency software lets several people be logged in at once and work on different parts of the business without any trouble. Two staff members viewing the same booking at the same time is perfectly normal. The care is needed when both try to edit the same record in the same moment, because the last change saved can replace the earlier one.

Many systems reduce that risk by locking a booking while someone has it open for editing, so the second person sees a message saying it is already in use and who has it. Others allow both to edit and simply record each change in the booking history, so you can see what was altered and by whom. Which behaviour you get depends on the software you are using and sometimes on your plan, so it is worth asking directly rather than assuming one or the other.

Actions that go out to a supplier, such as a cancellation or a date change, are a separate matter. Those are sent through the supplier system, and once one person has sent the request the second person cannot send it again. In that situation your staff member will normally get an error back from the supplier rather than a duplicate change, but the wording of that message is not always clear, which is why a quick refresh of the booking before acting saves confusion.

The practical answer is to give each staff member their own login rather than sharing one, so the record of who did what is accurate, and to agree internally on who owns a booking while it is being worked on. Ask us to confirm how editing and locking behave in your setup, and check whether your booking history shows the staff name against each change.