Crumble ground beef into a stockpot over medium-high heat; cook and stir until beef is crumbly, evenly browned, and no longer pink, 7 to 10 minutes. Drain and discard any excess grease.
Stir carrots, celery, and onion into the ground beef; cook and stir until vegetables are soft, about 5 minutes. Add potatoes, tomato sauce, crushed whole tomatoes, undrained peas, corn, and green beans to the pot; season with black pepper. Stir mixture until evenly mixed. Add thyme and bay leaf to the pot. Pour 1/4 cup water over the mixture if needed to moisten mixture, replenishing as needed while cooking.
Bring soup to a simmer, place a cover on the pot, and cook, stirring regularly, until potatoes are tender, about 1 hour.