Yes, this is normally possible, and it is one of the first things a corporate client asks for once they have been with you a while. The way it works is that the department or cost centre is captured at the moment of booking rather than worked out afterwards. You load the client's list of departments, cost centres or project codes into their account, and when their employee books a trip they pick one from a dropdown. That tag stays attached to the booking, and the travel spend reports can then be grouped by it.
Two things make the difference between a report your client trusts and one they ignore. Make the field compulsory, so a booking cannot be completed without it, and keep it as a dropdown rather than a free text box. If employees type the department name themselves you will end up with the same department spelled several different ways and the totals will never match what their finance team has. It is also worth giving the client their own login so they can pull the report whenever they like, instead of your staff being asked for it at the end of every month.
There are honest limits. The report only knows about what went through your portal, so anything the client books direct with an airline, or anything your staff took over the phone without entering the tag, will be missing until somebody adds it by hand. Extras the traveller settles at the hotel on checkout are not in your booking record at all, because that money never came through you. Whether the client gets their own reporting login, and whether reports can be emailed out on a schedule, depends on your plan.
Before you promise anything, ask the client for their department and cost centre list in the exact wording their finance team uses, and check which reporting options your plan includes.