directive.html

<form class="zip-input-group" zip-input ng-submit="checkZip()">
  <input class="zip-input-text" name="zip-text" type="text" maxlength="5" required>
  <input type="submit" value="Go!" class="zip-input-submit">
</form>

directive.js

angular.module('myApp')
    .directive('zipInput', function() {

        return {
            restrict: "A",
            scope: {},
            controller: [$scope, function zipInputController($scope) {
                
                $scope.checkZip = function() {
                    console.log("ZIP form submitted!");
                }
            }]
        }
});

Have you tried using the link prop instead of controller?

    .directive('zipInput', function() {

        return {
            restrict: "A",
            link: function ($scope, element, attrs) {
                $scope.checkZip = function() {
                    console.log("ZIP form submitted!");
                }
        }
});

I haven't tried it, and that may work, but I'm not a huge fan of putting all my logic inside the link function, when the controller is specifically designed to handle all that. I'm more interested in why it's not working. Thanks for the suggestion though!