Webform-CiviCRM integration: autocomplete employer
The anonymous user does not have permission to view contacts in your database. Be thankful for this default, it can save you from accidentally exposing your entire database to the world!

You can override this default by editing the "existing contact" field (from the webform tab) and unticking "Enforce Permissions". Typically you would also set one or more filters on that form (e.g. pick a group) to limit the autocomplete to just the organizations you want to expose.