Awesome Live Chat - Custom trigger

alc-trigger.js

(function($){
   $(document).ready(function($){
      $('.css-class').on('click', function(e){
         e.preventDefault();
         AwesomeLiveChat.vent.trigger('app:toggle');
      });
   })
})(jQuery);

Add the snippet to the javascript section in the plugin settings and replace .css-class with the css selector of a DOM element.