Preheat oven to 350°F. Line 12 cupcake molds with cupcake liners.
In a large bowl, combine the coconut oil, sugar, eggs, salt, and vanilla extract. Stir in rum, yogurt, lime zest and juice.
In a separate bowl, sift together flour and baking powder. Incorporate into wet ingredients.
Scoop batter into cupcake molds, filling 3/4 of the way up. Bake for 25-30 minutes, until the cupcakes are light golden brown and a knife comes out clean when inserted into the center of the cupcakes. Allow to cool completely before frosting.
To make the frosting, place all ingredients in a standing mixer with a whisk attachment and beat until light and fluffy, about 2 minutes. Begin with 1 tablespoon rum. If too thick, add in 1 more tablespoon!
Transfer frosting into a piping bag fitted with a large star tip. Frost cupcakes, then top with small mint leaves and a lime wedge!