Preheat the oven to 350 degrees F (175 degrees C). Line a 12-cup muffin tin with paper liners.

Stir coconut milk and coconut oil in a small skillet over low heat until liquid, 3 to 4 minutes. Remove from heat and transfer to a bowl. Stir in sugar, lemon juice, vanilla extract, and lemon extract until combined.

Mix together flour, baking soda, baking powder, and salt in a separate bowl. Gradually add flour mixture to coconut milk mixture and stir until thoroughly combined. Spoon batter into the prepared muffin cups, filling each 3/4 full.

Bake in the preheated oven until tops spring back when lightly pressed, 18 to 22 minutes. Cool in the tin for 5 minutes. Transfer to a wire rack and let cool, about 20 minutes.

Put icing in a plastic bag and snip off one of the bottom corners. Squeeze icing onto cooled cupcakes and sprinkle with coconut flakes and lemon peel.