AFAIK Starriott/Bonvoy, Hyatt, and Hilton all give elite benefits and elite nights/stay credit on government rates that are booked through their respective websites. If you book through some other website, including your employer's or agency's travel portal, there could be a problem, especially if it essentially books through hotels.com, etc.

I've noticed that these chains offer a variety of government rates, depending on the particular property and availability. Some have a government rate, which might or might not be available to contractors; you must look at the conditions for the particular rate at the specific hotel to know for sure, and you might want to print the rules at the time of booking. Other hotels have separate rate plans for federal government, state government (which might or might not be restricted to the state in which the hotel is located), local government, contractor, etc., plus sometimes military rates that can be better than any government rate.

Especially if a group are traveling together, note that if the room is direct billed, you generally get NO elite benefits or status credit or points for the stay. If there's a negotiated rate or contract for a (generally larger) group or meeting, what happens depends on the details in the contract, which *should* be explicit about this but generally isn't information that is distributed to individual travelers.