Heat the oven to 350°F (175°C). Prepare a 9x13-inch cake pan by greasing it and dusting it lightly with flour, or lining it with parchment paper for easy removal.
Combine the flour, cocoa powder, baking soda, and salt in a medium-sized bowl. Mix thoroughly and set aside.
Using a large mixing bowl, whisk together the brown sugar, vegetable oil, melted butter, and buttermilk until smooth and well blended.
Add the eggs and vanilla to the wet ingredients, mixing until evenly combined. Slowly incorporate the dry mixture into the wet mixture, stirring gently until just combined.
Gently fold in the shredded zucchini until it is evenly dispersed throughout the batter. Then stir in the chocolate chips.
Spread the batter evenly into the prepared pan. Bake for 35-40 minutes, or until a toothpick inserted in the center comes out clean or with a few moist crumbs clinging to it.
Once baked, remove the cake from the oven and allow it to cool completely in the pan on a wire rack before adding the frosting.