Heat oil over medium-high heat in a Dutch oven. Add chicken to the hot oil and brown on 1 side, about 5 minutes. Flip chicken; add carrots, white onion, red onion, bay leaves, garlic, black pepper, and salt. Mix until vegetables have softened and chicken has browned, about 8 minutes.
Pour red wine vinegar, white wine, and water into the Dutch oven. Sprinkle with achiote powder and cayenne; stir well. Reduce heat to medium. Cover and simmer until chicken is no longer pink in the center and an instant-read thermometer inserted into the center reads at least 165 degrees F (74 degrees C), about 15 minutes.