Preheat your oven to 350°F (175°C) and grease a 9-inch round cake pan.
In a large bowl, whisk together the yogurt, sugar, and vegetable oil until smooth.
Add the eggs and vanilla extract, and whisk until well combined.
In another bowl, combine the flour, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake in the preheated oven for 35 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan for 10 minutes before transferring to a wire rack to cool completely.