Preheat your oven to 350°F (175°C).
Grease and line a 9x13-inch baking pan with parchment paper.
In a large bowl, whisk together the flour, light brown sugar, baking powder, baking soda, cinnamon, ginger, and salt.
In another bowl, mix the plant-based milk, vegetable oil, dairy-free yogurt, apple cider vinegar, and vanilla extract until smooth.
Add the wet ingredients to the dry ingredients, mixing until just combined. Fold in the grated carrots.
Pour the batter into the prepared pan and spread it evenly.
Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool in the pan on a wire rack.