Preheat oven to 350 degrees F (175 degrees C). Lightly oil a 9-inch square pan.
Mix rice flour, sugar, potato starch, tapioca starch, cinnamon, baking powder, baking soda, xanthan gum, salt, and nutmeg together in a bowl. Add pineapple, 1/4 cup reserved pineapple juice, yogurt, carrots, walnuts, coconut, applesauce, vegetable oil, and vanilla extract to flour mixture; mix with a wooden spoon until well blended. Pour batter into the prepared pan.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, 30 to 35 minutes.
