Extension offices are usually affiliated with a state university, and sponsor the local 4-H group.
In Virginia, the extension office system is run through Virginia Tech, which provides most of the funding for extension offices in the counties. The counties normally supplement the funding the extension office gets from the State.
My guess is that each State program gets at least a portion of its funding from the USDA, but I am less familiar with the specifics of the federal part.
I don’t remember the Boy Scouts being funded by the Feds....