Preheat the oven to 350 degrees F (175 degrees C). Line a loaf pan with parchment paper.

Mix spelt flour, cinnamon, baking powder, salt, and baking soda together in a bowl. Mix soy milk, coconut oil, applesauce, coconut sugar, and vanilla extract together in another bowl. Fold flour mixture into the milk mixture until just combined. Add carrots; whisk well. Add walnuts and stir to evenly distribute.

Pour carrot cake batter into the loaf pan and even out the top with a spatula.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 35 to 40 minutes. Let cool in the pan for 5 minutes. Turn cake out onto a cooling rack.

Beat confectioners' sugar, vegan butter, and soy milk together in a bowl using an electric mixer to make the icing. Add lemon juice, vinegar, and vanilla extract. Mix well. Frost cake once cool.