Begin by preheating your oven to 375°F (190°C). Line a 12-cup muffin tin with cupcake liners or lightly grease the cups to prevent sticking.
In a large mixing bowl, whisk together the vegetable oil, applesauce, raw cane sugar, eggs, and vanilla extract until well combined.
In a separate bowl, mix the all-purpose flour, baking powder, baking soda, salt, and ground cinnamon. Stir until the dry ingredients are evenly distributed.
Gradually add the dry ingredients to the wet mixture, stirring just until combined. Be careful not to overmix, as this can lead to dense muffins.
Gently fold in the grated carrots, grated zucchini, and chopped walnuts. Ensure the mixture is evenly distributed without overworking the batter.
Spoon the batter evenly into the prepared muffin tin, filling each cup about two-thirds full. This allows room for the muffins to rise without overflowing.
Place the muffin tin in the preheated oven and bake for 20-25 minutes. Check for doneness by inserting a toothpick into the center of a muffin; it should come out clean when the muffins are ready.
Remove the muffin tin from the oven and let the muffins sit in the tin for about 10 minutes. Then, transfer them to a wire rack to cool completely.