Allowing anonymous access to Google Apps Script web app (Google Apps account)

Have you checked with your Google Apps Administrator? They might have enforced a policy that one can not share the docs outside the organization.

Here is the documentation on docs sharing policy. https://support.google.com/a/answer/60781?hl=en

This setting is controlled by the domain administrator via the "Allow users in [domain] to publish files on the web or make them visible to the world as public or unlisted files" options that's under "Settings for Drive and Docs".

Note that after checking that option, you'll have to create a new Script in order to be able to publish it with "Anonymous" access. Existing scripts don't seem to get that option even after the setting has been changed.

Also note the Google issue tracker for this: https://issuetracker.google.com/issues/36764781