Preheat the oven to 350°F (175°C). Grease and flour a 9×13-inch baking pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, and lemon zest. Set aside.
In a large bowl, beat the granulated sugar, light brown sugar, and vegetable oil together until well combined.
Add the eggs one at a time to the sugar mixture, beating well after each addition.
Mix in the fresh lemon juice and vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined.
Fold in the grated zucchini until evenly distributed throughout the batter.
Pour the batter into the prepared baking pan and spread it evenly.
Bake 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 completely in the pan on a wire rack.
While the cake cools, prepare the cream cheese frosting. In a large bowl, beat the softened cream cheese and unsalted butter together until smooth and creamy.
Gradually add the powdered sugar, one cup at a time, beating well after each addition.
Add the fresh lemon juice and vanilla extract and beat until the frosting is smooth and fluffy.
Once the cake is completely cool, spread the cream cheese frosting evenly over the top.
Slice and serve. Store any leftovers covered in the refrigerator.