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Preheat oven to 350°F. Prepare cake mix as directed on package, adding orange extract. Bake as directed on package for cupcakes. Cool cupcakes on wire rack.
Beat butter and vanilla in large bowl until light and fluffy. Gradually add confectioners' sugar, beating well after each addition and scraping sides and bottom of bowl frequently. Add milk; beat until light and fluffy. Remove 1/2 of the frosting to medium bowl. Tint orange with yellow and red food colors. Add cocoa powder to remaining frosting; beat until well blended, adding additional milk as needed.
Make an indentation in the center of each cupcake using the pointed end of a sugar ice cream cone or handle of a wooden spoon, making sure not to break through bottom of cupcake. (The indentation should be pointed at the bottom and a little wider at the top – like a carrot.) Spoon orange frosting into resealable plastic bag. Cut a small piece off one of the bottom corners of bag. Pipe orange frosting into cupcakes to fill indentation. (This is the carrot.) Spread cupcakes with chocolate frosting, keeping top of the carrot visible. Place string licorice on top of each carrot to resemble the green tops of the carrot.