Pour water and bulgur into a saucepan. Bring to a boil; cover and simmer until tender, 12 to 15 minutes. Drain off excess water.
Heat olive oil in a large pot over medium heat. Add carrots, onions, green bell pepper, red bell pepper, and jalapeno pepper; cook and stir until softened and starting to brown, 5 to 10 minutes. Stir in garlic; cook and stir until fragrant, 1 to 2 minutes.
Stir cooked bulgur, tomato juice, kidney beans, black beans, diced tomatoes, tomato sauce, chili powder, taco seasoning, cumin, cayenne, and pepper into the pot. Bring to a boil; reduce heat and simmer until flavors combine, 20 to 30 minutes.