Preheat oven to 350 degrees. Brush two (6 each) donut pans with some oil.
In a large mixing bowl, whisk together the brown rice flour, tapioca starch, cocoa powder, baking soda, and salt, breaking up any lumps from the cocoa powder.
In another mixing bowl, whisk together the sugar, rice milk, oil, and vanilla. Add the wet ingredients to the dry and mix well.
Spoon the batter into the prepared pans, filling the donut cavities almost full. Alternatively, you can put the batter into a large food storage bag, cut off about ¼ inch from one of the corners and squeeze the mixture into the pans. Bake for 15 minutes or until the donuts spring back when touched lightly and a tooth pick inserted into the center comes out clean. Let cool in the pans for 10 minutes, then move to a wire rack to cool completely.