Preheat oven to 350 degrees F (175 degrees C). Line 2 baking sheets with parchment paper.
Mix ground flax seed and water in a bowl; let sit until thickened, about 10 minutes.
Cream vegan margarine and coconut sugar together until smooth, about 3 minutes. Add flax mixture and vanilla extract; mix until well combined. Stir in flour, baking powder, and salt; mix well until a soft dough forms. Wrap dough in plastic wrap and chill for 30 minutes.
Roll out dough to 1/4 to 1/2-inch thickness on a floured surface. Dip cookie cutter in flour, cut out cookies, and place them on the prepared baking sheets.
Bake in the preheated oven until lightly golden, about 10 minutes. Chill completely before adding frosting.
Combine confectioners' sugar, soy milk, vanilla extract, and food coloring in a bowl. Dip cookies in frosting and place on a wire rack until frosting is set, about 30 minutes.