Preheat the oven to 350°F. Spray an 8x8 square baking dish or pan and set it aside.
Zest and juice the lemons.
Add the sugar and lemon zest to a large mixing bowl. Rub the zest into the sugar with your fingers until it becomes fragrant.
In a separate small or medium bowl, whisk together the flour, baking soda, and salt. Pour the flour mixture into the large bowl with the sugar, and whisk until thoroughly mixed.
Pour the oil and lemon juice into the dry ingredients and slowly whisk until the batter is too thick to stir.
Add the vinegar to the water, and give it a quick stir. Pour that into the batter, and carefully whisk until the batter is wet and starts to bubble. That's the vinegar, lemon juice, and baking soda reacting!
Pour the batter into the greased 8x8 square baking dish, and bake the cake for 35-40 minutes or until the cake browns on the edges, and a toothpick inserted in the middle of the cake comes out clean. Let the cake cool for 30 minutes to an hour.
After the cake has cooled completely, make the lemon icing. Add the powdered sugar to a small bowl. Whisk in the lemon juice and salt until everything is combined, and a smooth icing has formed.
Pour the icing over the cake, and sprinkle some lemon zest on top, if desired, while the icing is still wet. Let the cake rest for 30 minutes so the icing will set slightly, then serve at room temperature.