Preheat the oven to 300 degrees F (150 degrees C). Line the bottom of a 9-inch springform pan with parchment paper.
Stir gingersnap crumbs and melted margarine in a bowl until well combined. Press crumb mixture into the bottom of the prepared pan and place in the freezer.
Beat chèvre and sugar with an electric mixer in a large bowl until smooth. Add eggs, one at a time, beating until incorporated after each addition. Mix in coconut milk. Mix in coconut yogurt, flour, and vanilla just until incorporated. Pour batter into chilled crust.
Bake in the preheated oven for 1 hour. Turn the oven off, keep the door closed, and allow cheesecake to cool inside for at least 2 hours. Refrigerate for an additional 4 to 6 hours before serving.