Preheat your oven to 350°F (175°C). Grease and line a round cake pan (8-inch diameter).
In a large bowl, whisk together flour, brown sugar, baking soda, baking powder, cinnamon, ginger, and salt.
In another bowl, whisk together non-dairy yogurt, vegetable oil, non-dairy buttermilk, and vanilla extract.
Pour the wet ingredients into the dry ingredients and mix until just combined. Fold in the grated carrots.
Pour the batter into the prepared cake pan and spread evenly. Bake for 25-30 minutes or until a toothpick inserted into the center comes out clean.
Let the cake cool completely in the pan.