Preheat the oven to 350°F (175°C). Grease and flour a 9×5-inch loaf 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 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.
Alternately add the dry ingredients and sour cream to the wet ingredients, beginning and ending with the dry ingredients. Mix gently until just combined.
Fold in the semi-sweet chocolate chips.
Pour the batter into the prepared loaf pan and smooth the top with a spatula.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean or with just 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.