Sift flour, baking powder, and salt together in a medium bowl; set aside.
Beat sugar and margarine in a large bowl with an electric mixer until smooth. Wrap dough; chill in the refrigerator for 2 hours.
Beat first egg into butter mixture. Beat second egg into mixture along with vanilla extract; add flour mixture and stir until dough is just combined.
Wrap dough; chill in the refrigerator for 2 hours.
Preheat the oven to 400 degrees F (200 degrees C). Grease two cookie sheets.
Roll out dough on a clean floured surface to 1/4-inch thickness. Cut out shapes using cookie cutters; arrange them on the prepared cookie sheets. Gather scraps and repeat this step.
Bake in batches in the preheated oven until edges are golden, about 6 to 8 minutes. Cool on the baking sheet briefly before removing to a wire rack to cool completely.