Preheat oven to 350°F. Grease a 9x13-inch cake pan.
Whisk gluten-free flour, sugar, cocoa powder, baking soda, and salt in a large mixing bowl. Add water, oil, and vinegar. Mix until a smooth batter forms, about 30 seconds. Let the batter sit for 5 minutes to thicken.
Spread batter evenly into the pan.
Bake until a cake tester inserted into the center of the cake comes out clean, about 35 minutes.
Note 1: The recipe was tested with Bob's Red Mill 1:1 Gluten-Free Baking Flour. Using a different gluten-free flour can change the results.
Note 2: To make the frosting dairy-free, use a dairy-free buttery spread and dairy-free milk.