On a Drupal user application form, is it possible for the user to lookup existing users?
Yes it is possible but would require some custom code.

Assuming you already have a form element where your autocomplete search would take place, in a hook_form_alter() or hook_form_FORM_ID_alter() you could use something like:

  // Add ajax autocomplete.
  $form['name_of_your_text_element']['#attached']['js'][] = 'misc/autocomplete.js';
  $form['name_of_your_text_element']['#autocomplete_path'] = 'ctools/autocomplete/user';


Note that you will probably need to adjust some permissions in order to allow anonymous users to see your users' information. Check /admin/people/permissions if you need to fine-tune something there.

You also need to have the ctools module enabled in order to use this functionality.