Preheat oven to 350°F. Grease two 8-inch round cake pans and line the bottoms with parchment paper circles.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt.
In a large bowl, whisk together the brown sugar, non-dairy milk, melted non-dairy butter, white vinegar, and vanilla extract until combined.
Add the dry ingredients to the wet ingredients and whisk until a smooth batter forms.
Divide the batter evenly between the two prepared pans. Sprinkle 3 tablespoons of mini chocolate chips over each pan and use a butter knife to swirl them into the batter.
Bake for 28–31 minutes, or until a toothpick inserted into the center of each cake comes out clean.
Cool in pans for 30 minutes, then turn out onto a parchment-lined baking sheet and cool completely before assembling.