Prepare your camp oven or Dutch oven. If using a campfire, build a good bed of coals. For a camp stove, preheat a covered cast iron Dutch oven over low heat.
In a large bowl, whisk together the flour, granulated sugar, baking powder, baking soda, and salt. Make a well in the center.
In a separate bowl or large measuring cup, whisk the buttermilk, oil, egg, and vanilla until smooth.
Pour the wet ingredients into the well of the dry ingredients. Stir with a wooden spoon just until combined. A few streaks of flour are fine — do not overmix.
Gently fold in the blackberries with a rubber spatula. Use 4-5 strokes only. Overmixing will turn the batter gray.
Grease a 12-cup muffin pan or line with paper liners. Divide the batter evenly among the cups. Sprinkle turbinado sugar on top of each muffin.
Place the muffin pan into the preheated Dutch oven. Set the lid on. For a campfire, arrange 6-8 hot coals on the lid and 4-5 underneath. For a camp stove, cook over low heat with the lid on.
Bake for 18-22 minutes, rotating the Dutch oven halfway through by a quarter turn. Muffins are done when a toothpick inserted in the center comes out clean or with a few moist crumbs.
Carefully remove the muffin pan using heavy oven mitts. Let the muffins cool in the pan for 5 minutes, then transfer to a wire rack. Serve warm or at room temperature.