Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking pan.
Mix milk and vinegar together in a small bowl. Let sit until curdled, about 5 minutes.
Mix flour, sugar, ground flax seeds, cocoa powder, baking soda, baking powder, and salt together in a bowl. Make a well in the center of the flour mixture; add milk-vinegar mixture, coffee, eggs, and vegetable oil. Beat with an electric mixer until batter is well mixed.
Pour batter into the prepared baking pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 30 minutes.