Preheat oven to 350 degrees and line a muffin pan with cupcake liners.
Whisk together the flour, baking soda, baking powder, salt, and cinnamon. Set aside.
In a large bowl, beat together the vegetable oil and sugar until well combined. Add the applesauce and vanilla and mix until combined.
Add the dry ingredients to the oil mixture and mix until smooth.
Fold in the grated carrots and chopped walnuts.
Fill each cupcake liner 2/3 full of batter. Bake in the preheated oven for 25 to 30 minutes, until a toothpick inserted in the center comes out clean.
To make the icing, beat together the vegan butter, vegan cream cheese, and vanilla extract until smooth. Add the powdered sugar and beat until combined.
When the cupcakes are cooled completely, pipe the icing on top and garnish with grated carrots, shredded coconut, and walnuts.