Preheat the oven to 350 degrees F. Grease an 8 1/2 by 4 1/4 by 2 1/2-inch loaf pan. Line the bottom with parchment paper. Grease and flour the pan.
Sift together the flour, baking powder, and salt into 1 bowl. In another bowl, whisk together the yogurt, 1 cup sugar, the eggs, lemon zest, and vanilla. Slowly whisk the dry ingredients into the wet ingredients. With a rubber spatula, fold the vegetable oil into the batter, making sure it's all incorporated. Pour the batter into the prepared pan and bake for about 50 minutes, or until a cake tester placed in the center of the loaf comes out clean.
Meanwhile, cook the 1/3 cup lemon juice and remaining 1/3 cup sugar in a small pan until the sugar dissolves and the mixture is clear. Set aside.
When the cake is done, allow it to cool in the pan for 10 minutes. Carefully place on a baking rack over a sheet pan. While the cake is still warm, pour the lemon-sugar mixture over the cake and allow it to soak inches Cool.
For the glaze, combine the confectioners' sugar and lemon juice and pour over the cake.