Break ground beef into small pieces into a large pot over medium heat; cook and stir until beginning to brown, 3 to 5 minutes. Add onion, green bell pepper, yellow bell pepper, and carrot to ground beef; cook and stir until the vegetables begin to soften, 5 to 7 minutes. Stir zucchini and yellow squash into the beef mixture; continue cooking until the beef is completely browned, 3 to 5 minutes more. Drain excess liquid from the pot and return to medium heat.
Stir crushed tomatoes, kidney beans, black beans, diced tomatoes, corn, and garlic into the beef mixture; bring to a boil. Season mixture with chili powder, brown sugar, paprika, oregano, cayenne pepper, cumin, salt, and black pepper.
Reduce heat to low, place a cover on the pot, and cook chili at a simmer for 30 minutes.
