In an ungreased glass or ceramic 8-inch square pan (do not use metal), add the flour, sugar, cocoa powder, baking soda and salt, and stir to combine with a fork.
Make three wells in the mixture and to those wells add the vinegar, vanilla extract and vegetable oil. Pour the water over the entire mixture and mix with a fork until thoroughly combined and no dry ingredients remain.
Bake until a toothpick inserted in the center comes out clean, 26 to 30 minutes. Place the pan on a wire rack to cool completely before frosting.
Make the Vanilla Frosting: Using the wire whisk attachment of your stand mixer, whip the butter on medium-high speed for 5 minutes, stopping to scrape the bowl once or twice. Reduce the speed to low and gradually add the powdered sugar. Once all of the powdered sugar is incorporated, increase the speed to medium-high and add the vanilla, mixing until incorporated. Whip at medium-high speed until light and fluffy, about 2 minutes, scraping the bowl as needed.
Using a spatula, spread the vanilla frosting evenly over the top of the cake. The cake can be stored, covered with plastic wrap, at room temperature, for up to 4 days.