Preheat oven to 350 degrees. Line two baking sheets with parchment paper or silicon baking mats. Remove non-dairy margarine from the refrigerator and let it sit while you are mixing the dry ingredients.
Prepare the pumpkin puree: Place a paper towel in a bowl, add the pumpkin puree and let it sit until you are ready to use. This will help soak up the extra moisture.
Make the dough: mix the flour, baking soda, cinnamon, pumpkin pie spice and salt in a medium bowl and set aside.
Using a mixer with a paddle attachment, cream the non-dairy margarine and the sugars. Add the applesauce and vanilla and pumpkin puree and mix until combined. Slowly add the dry ingredients and mix until well combined and creamy. Then stir in the old-fashioned oats until just combined.
Using a medium cookie scoop, place cookie dough on prepared cookie sheets. Flatten the dough slightly before placing in the oven. Bake for 12-14 minutes. Remove form the oven and allow to cool for 10 minutes before icing.
Make the icing: Melt the coconut oil and let cool. In a small bowl, mix the confectioner’s sugar, non-dairy milk and vanilla. Slowly stir in the coconut oil and stir until ingredients are thoroughly combined. Either dip the cookie into icing mixture or drizzle icing on top.