Preheat oven to 350°F (175°C) and grease a 9x9 inch baking pan.
Prepare flax eggs by mixing ground flaxseeds with water and letting sit for 5 minutes until thickened.
In a large bowl, whisk together flour, sugars, baking soda, baking powder, salt, and spices.
In a separate bowl, combine pumpkin puree, oil, flax eggs, and vanilla extract.
Add the wet ingredients to the dry ingredients and stir until just combined.
Pour batter into the prepared pan and smooth the top.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Allow cake to cool completely in the pan on a wire rack before serving.