Preheat oven to 350 degrees F (175 degrees C). Line 2 baking sheets with parchment paper.
Sift flour, baking powder, cinnamon, baking soda, ginger, allspice, and salt into a bowl.
Cream coconut oil, molasses, and sugar in a bowl with an electric mixer; add vanilla extract. Stir in flour mixture; mix until a sticky dough forms, about 2 minutes. Wrap dough in plastic wrap and chill for 2 hours.
Roll out dough on a floured surface to 1/4 to 1/2-inch thickness. Dip cookie cutter in flour, cut out cookies, and place on the prepared baking sheets.
Bake in the preheated oven until lightly golden, 8 to 10 minutes.