Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line it with parchment paper.
In a medium bowl, sift together the all-purpose flour, unsweetened cocoa powder, baking soda, baking powder, and salt. Set aside.
In a large bowl, whisk together the granulated sugar, light brown sugar, and vegetable oil until well combined.
Add the eggs one at a time to the sugar and oil mixture, beating well after each addition.
Stir in the vanilla extract.
Add the grated zucchini, mashed banana, and sour cream to the wet ingredients and mix until combined.
Gradually add the dry ingredients to the wet ingredients, folding gently until just combined. Do not overmix.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 40 to 45 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 10 minutes.
Transfer the cake to a wire rack to cool completely before serving.