Place vanilla cookies in a food processor and process into fine crumbs.
Heat chocolate chips and corn syrup together in a saucepan over low heat. Cook, stirring often, until chocolate is melted and smooth, about 5 minutes.
Remove from heat and stir in rum and confectioners' sugar until smooth. Fold in cookie crumbs; dough will be sticky.
Place saucepan in the refrigerator until dough is firm and easy to roll, about 15 minutes. Cover 2 plates with waxed paper; dust with confectioners' sugar.
Roll dough into 1-inch balls and place on the prepared plates. Dust rum balls with confectioners' sugar. Refrigerate until firm, about 30 minutes.
Remove rum balls from refrigerator and transfer to a resealable bag, including the extra confectioners' sugar. Seal the bag and shake to coat the rum balls completely with confectioners' sugar.
