In the US context, it sounds like you're asking about 42 CFR 52.203-5 ("Covenant Against Contingent Fees"). In that context, contingent fees means something close to payment of a commission to someone working as a broker, salesperson, negotiator or in a similar role.
The rule applies to contracts under 41 USC §254(a) as well as bid procurements regulated by FAR §§ 3.400 and 3.403. There are exceptions to the prohibition, including for external agents, but even where it is applies it would not forbid simple use of a 'middleman' — it just prohibits that agent from being compensated by commission on the deal. The idea is to prevent agents with "improper influence" (or claimed influence) from being paid by commission, or having their fixed payment conditioned on closing the deal.
This publication from Venable provides more details on the specifics and the exceptions.