Oddly enough, the program is called "AOC," for Architect of the Capitol. Flag request is accessed via links provided for U.S. Reps and Senators on the web page. Each of the congress critter websites has a link to the Flag Request form and instructions for completing the request form and payment. Flags flown over the Capitol take substantially longer to receive than those not raised up the flagpole.
https://www.aoc.gov/flags