You can publish a Google Apps Script as a web app and anyone can run your scripts directly from their browsers. For instance, if you have a file upload Google Form, you can publish the script as a public web app and even anonymous users, who are not logged into their Google accounts, will be able to upload files to your Google Drive.

When deploying a Google Apps Script ha a web app, you can specify who has access to your web app:

1. Only myself - only the script owner can access the web.
2. Any member of domain - only members who are part of your Google Apps domain can access the app.
3. Anyone - Anyone can access the web app, but they need to be logged into their own Google account.
4. Anyone, including anonymous - Users who do not have Google accounts can also access your web app.

If you are unable to publish your web app for anonymous usage, it is likely that your Google Apps admin has disabled the option for the organization.

Allow users to share outside Google Apps domain

Some Google Apps admins, for privacy reasons, may disallow users in the domain from sharing documents and files outside the organization. In such cases, the option to publish a web app for anonymous usage is unavailable too.

1. Sign in to the Google Admin console at admin.google.com
2. Go to Apps > Google Apps > Drive > Sharing settings.
3. Under Sharing outside of organization, select ON.
4. Select the option - Allow users in organization to publish files on the web or make them visible to the world as public or unlisted files.

You can use the Google Drive Auditor to audit the sharing permissions of your files and know who has access to Drive.