Question

Is there a way to access the widgets within a panel in GAS?

Something like:

function clickHandler_(e) {
  var app = UiApp.getActiveApplication();
  var panel = app.getElementById(e.parameter.whatever); // short-cutting here
  for (var i=0; i<panel.widgets.length; i++) {
    var widget = panel.widgets[i];
    // do something with them
  }
  return app;
}

Answer 1

There isn’t something simple like this.
You have to give all widgets an id when adding them and save them somewhere so you can retrieve it later. For example, as a tag on the panel:

function doGet(e) {
  var app = UiApp.createApplication();
  ...
  var panel = app.createXYZPanel().setId('mypanel');
  var IDs = ''; //list of child widgets on this panel
  panel.add(widget.setId('id1'));
  IDs += ',id1';
  panel.add(widget2.setId('id2'));
  IDs += ',id2';
  //and so on...
  panel.setTag(IDs); //save the IDs list as a tag on the panel
  ...
  return app;
}

//...later on a handler
function handler(e) {
  var app = UiApp.getActiveApplication();
  //the panel or a parent must be added as callback of the handler
  var IDs = e.parameter.mypanel_tag.split(',');
  for( var i = 1; i < IDs.length; ++i ) { //skipping the 1st empty element
    var widget = app.getElementById(IDs[i]);
    //do something
  }
  return app;
}

By the way, you probably know that, but your code had a mistake:
It’s not UiApp.getElementById, but app.getElementById.