Preheat your oven to 350°F (175°C).
Grease and line a 9x13-inch baking pan with parchment paper.
In a large bowl, whisk together the flour, granulated sugar, light brown sugar, cocoa powder, baking powder, baking soda, and salt.
In another bowl, mix the plant-based milk, vegetable oil, dairy-free yogurt, apple cider vinegar, and vanilla extract until smooth.
Add the wet ingredients to the dry ingredients, mixing until just combined. Do not overmix.
Pour the batter into the prepared pan and spread it evenly.
Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool completely in the pan on a wire rack.