On a Drupal user application form, is it possible for the user to lookup existing users? I'm looking to add a user lookup form to my Drupal site's application form. One question would ask the applicant/registrant to look up an already existing user on the site. What's the best way to go about something like this in Drupal?
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.