Preheat your oven to 350°F (175°C). Grease and flour a 9x13 inch baking pan.
In a large bowl, whisk together the flour, cocoa powder, baking powder, baking soda, and salt.
In another bowl, mix the vegetable oil, sugar, eggs, and vanilla extract until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Fold in the shredded zucchini.
Pour the batter into the prepared baking pan and spread it out evenly.
Bake for 45-50 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool completely in the pan.
For the frosting, beat the cream cheese and butter together until smooth. Gradually add the powdered sugar and vanilla extract, beating until creamy.
Spread the frosting over the cooled cake. Slice and serve.