Preheat the oven to 350 degrees F (175 degrees C). Spray an 8 1/2x4 1/2-inch loaf pan with nonstick cooking spray.
Whisk together oat milk and ground flax seeds in a small bowl until well combined, then place in the refrigerator for 15 minutes to set.
Shred zucchini and gently press out moisture. Set aside.
Beat the oat milk/flax seed mixture, brown sugar, oil, honey, and vanilla extract in a large bowl with a hand mixer until smooth. Add flour, cocoa powder, salt, baking soda, and baking powder, and mix until well combined. Stir in zucchini. Pour batter into the prepared loaf pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 55 to 65 minutes. Cool in the pan for 15 minutes before removing the bread from the pan to cool on a wire rack completely before slicing.