Preheat the oven to 350ºF. In a large bowl, stir together the flour, sugar, salt, baking soda, and cocoa powder until well combined.
Add 1 cup water to a liquid measuring cup, then add the vanilla extract and vinegar to the water.
Add the oil to the bowl of dry ingredients, followed by the water mixture. Stir until the chocolate cake batter is mostly smooth. Make sure no dry flour remains on the bottom of the bowl.
Pour the cake batter into an 8×8″ or 9×9″ baking dish. Transfer the baking dish to the oven and bake the cake for 35 minutes.
If using the chocolate icing, let the cake cool for at least an hour after baking before adding the icing.
Wait until the cake is cool, then prepare the icing. Add the powdered sugar, cocoa powder, and vanilla extract to a bowl. Begin adding water, 1 tablespoon at a time, until it forms a thick but pourable icing (about 3 Tbsp total). If you let the icing sit, it may begin to dry, but you can add a splash more water to make it moist again.
Pour the icing over the cooled cake and spread until the cake is evenly covered. Slice the cake into 9 pieces and serve.