Preheat the oven to 350°F. Lightly grease an 8" or 9" round or square pan that’s at least 2" deep; line with parchment or a reusable pan liner.
To make the cake: In a medium bowl, whisk together the flour, baking powder, salt, and baking soda. Set aside.
In a separate bowl or large measuring cup, whisk together the water, sugar, vegetable oil, lemon juice, lemon zest or extract, and vanilla. Stir to dissolve the sugar.
Add the wet ingredients to the dry and stir to combine. It's OK if a few small lumps remain.
Pour the batter into the prepared pan.
Bake the cake for 30 to 35 minutes, until the top feels set, the edges begin to pull away from the sides of the pan, and a toothpick inserted into the center comes out clean. Once the cake is cool enough to handle, turn out onto a wire rack to cool completely.
To make the frosting: In a medium bowl, beat together all of the frosting ingredients, starting with the lower amount of water. Mix until the frosting is smooth and silky. Adjust with additional water as needed. Spread over the cooled cake.
Garnish the cake with candied lemon peel (see “tips,” below) or fresh berries, if desired.
Storage information: Store the vegan lemon cake, covered, at room temperature for several days; freeze for longer storage.
