Preheat oven to 350 degrees F (175 degrees C). Line 2 baking sheets with parchment paper.
Sift flour, baking soda, and salt into a bowl.
Cream sugar and vegan margarine with an electric mixer in a bowl, about 2 minutes; add vanilla extract. Stir in flour mixture. Add coconut milk, 1 tablespoon at a time, until dough comes together. Wrap in plastic wrap and chill for 30 minutes.
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 them on the prepared baking sheets.
Bake in the preheated oven until lightly golden, about 10 minutes.