Combine 4 tomatoes, 1/2 an onion, and garlic in a blender. Blend until smooth.
Heat oil in a soup pot over medium heat. Add the blended tomato mixture; cook until mixture changes color, about 5 minutes. Add chicken stock, reduce heat, and simmer while you prepare the meatballs.
Place ground chicken in a bowl. Add 3 chopped tomatoes, chopped onion, egg, rice, mint, oregano, salt, and pepper. Mix well with your hands until mixture is uniform. Take about 1/4 cup of mixture and form into round meatballs using wet hands. Place on a baking sheet. Repeat with remaining chicken mixture.
Bring stock mixture to a boil. Place a meatball into a ladle and submerge into the boiling stock. Remove ladle and repeat with remaining meatballs.
Place zucchini, chayote, and carrots into the stock. Bring to a boil again. Lower heat, season with salt, and cover. Simmer until meatballs are no longer pink in the center and vegetables are soft, about 30 minutes.