Preheat oven to 350°F (175°C). Grease and line an 8-inch round cake pan.
In a bowl, whisk flour, baking powder, baking soda, and salt.
In a separate bowl, whisk eggs and sugar until pale and slightly thickened.
Slowly whisk in olive oil until smooth.
Add yogurt, lemon zest, lemon juice, and vanilla; whisk just until combined.
Fold dry ingredients into wet ingredients until just incorporated—do not overmix.
Pour batter into prepared pan and smooth the top.
Bake 35–40 minutes, until the center springs back and a toothpick comes out clean.
Cool 10 minutes in the pan, then transfer to a rack. Dust lightly with powdered sugar if desired.