Combine vegetable juice and next 6 ingredients (juice through garlic) in a small saucepan; bring to a boil. Reduce heat, and simmer 5 minutes, stirring constantly. Remove from heat.
Place the chicken on a grill rack coated with cooking spray. Grill for 30 minutes or until done, turning frequently. Baste the chicken frequently with sauce after 15 minutes. Yield 4 servings (serving size: 2 drumsticks).