Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13x2-inch baking dish.

Whisk flour, sugar, baking soda, ground cinnamon, salt, and nutmeg together in a bowl. Beat carrots, mandarin oranges, oil, eggs, vanilla extract, and orange zest into flour mixture with an electric mixer until batter is smooth, about 2 minutes. Pour batter into prepared baking dish.

Bake in the preheated oven until a toothpick inserted into the center of the cake comes out clean, 40 to 50 minutes.