We were in DC last week. It looked to us like there were a number of food courts either a walk or a short metro ride from the mall. We ate at the food court in the Federal Triangle building, which we walked to from a Smithsonian. Union Station, L’Enfant Plaza looked to have a similar set up. Also, ask! We asked security guards at the Madison building and they were super helpful in pointing us in the direction of restaurants-they tend to be clumped together.
Also, we had food with us every day for lunch. We picknicked near the Washington Monument one day, in the zoo another, and outside the museums another. Also, we had no problem taking the packed food bags in to buildings, including museums, and we had food for the picky eaters and supplements (fruit) for all of us and then just purchased a few items at the museum cafes.