Preheat the oven to 350 degrees and line a 12-cup muffin tin with paper cup liners or silicone muffin liners.
Pour 1 tablespoon of ACV into a small liquid measuring cup, and then add coconut milk until you hit the ½ cup mark. This is your buttermilk for the cupcakes. Set this a side to add later. You can also start melting your coconut oil here.
In a medium mixing bowl, blend the eggs and coconut sugar using hand beaters until smooth.
With the hand mixer still running, pour in the melted coconut oil and blend until smooth. Add the coconut milk/ACV mixture to the wet ingredients, and blend until combined.
Add the rest of the ingredients, including the carrots, and use your hand mixer to blend on low until the batter is combined well.
Scoop the cupcake batter into your muffin pan, and bake the cupcakes at 350 degrees for 22 minutes, until the tops spring back to your touch, and a toothpick comes out clean. Let the cupcakes cool in the pan for 5 minutes before transferring to a cooling rack to cool completely before frosting.