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, whisk together the all-purpose flour, unsweetened cocoa powder, baking soda, baking powder, and salt until well combined.
In a large bowl, whisk together the granulated sugar, light brown sugar, and vegetable oil until smooth.
Add the eggs one at a time to the sugar and oil mixture, beating well after each addition.
Stir in the vanilla extract.
Fold in the grated zucchini and mashed banana until evenly distributed.
Add the dry ingredients to the wet ingredients in batches, alternating with the sour cream, beginning and ending with the dry ingredients. Mix gently until just combined; do not overmix.
Fold in the semisweet chocolate chips.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 38 to 42 minutes, or until a toothpick inserted into the center comes out clean or with a few moist crumbs.
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 slicing and serving.