Combine first 7 ingredients, stirring with a fork until mixture resembles coarse meal; sprinkle 1 1/2 tablespoons achiote mixture evenly over beef, tossing to coat. Set remaining achiote mixture aside.
Heat oil in Dutch oven over medium-high heat. Add beef mixture; cook 2 minutes, browning on all sides. Remove beef from pan. Add onion to pan; sauté 3 minutes. Add remaining achiote mixture; cook 2 minutes, stirring frequently. Return beef mixture to pan. Stir in broth and tomatoes; bring to a boil. Cover, reduce heat, and simmer 35 minutes. Add potato, chayote, and carrot; cover and simmer 35 minutes or until potato is tender. Remove from heat. Stir in vinegar.
When handling achiote paste, wear gloves to avoid staining your hands. If you can't find chayote, use summer squash instead.