In a Dutch oven or 5-quart saucepan with a lid, heat oil over medium. Add onion and garlic and cook, stirring occasionally, until tender, 5 to 7 minutes. Add squash, bell pepper, and chile powder; season with salt and pepper and cook, stirring 1 minute. Add 1/2 cup water. Cover and simmer until vegetables are crisp-tender, about 7 minutes.
Stir in tomatoes and their juice, breaking them up with a spoon; add chickpeas, black beans, 1/4 cup cilantro, and 1/2 cup water. Bring to a boil. Reduce to a simmer, partially cover, and cook until lightly thickened, about 20 minutes.
Season with salt and pepper. Stir in remaining cilantro and spoon into serving bowls. Serve with lime wedges.