Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg.
In a large bowl, beat the granulated sugar, light brown sugar, and eggs until smooth and well combined.
Add the vegetable oil and vanilla extract to the sugar and egg mixture and mix until incorporated.
Stir the dry ingredients into the wet ingredients until just combined.
Fold in the grated zucchini, grated carrot, and grated apple until evenly distributed.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 15 minutes.
Transfer the cake to a wire rack to cool completely before serving.