Preheat oven to 375°F. Place cupcake liners into muffin pan. Grate carrot and drain pineapple; set aside.
Stir together sugar, oil, and applesauce. Add eggs and vanilla and stir until well combined.
In a separate bowl, combine flours, baking powder, baking soda, cinnamon, and salt.
Add dry ingredients to sugar mixture and stir just until combined.
Fold in carrot and pineapple, then spoon into prepared muffin tin, filling a little more than 3/4 full.
Bake for 20 minutes or until a toothpick inserted into the center of a muffin comes out clean.
Cool in pan for 5-10 minutes, then remove to a rack to finish cooling.