Mix flour, chicken broth, and water in another bowl.
Heat oil in a large skillet or wok over high heat; cook and stir peppers, carrots, broccoli, and mushrooms until just tender, about 3 minutes.
Toss vegetables with soy sauce mixture; cook and stir for 1 minute. Gradually stir flour mixture into vegetables; bring to a boil and cook until thickened, about 3 minutes.