Project URL: https://gomix.com/#!/project/puny-hygienic
We run a public slack team for an open source project. The service that we previously used to handle invites was shut down, so we moved it to gomix instead. It is a very simple app that basically makes an HTTP POST call using a private key. You can use the gomix page directly or call it via a form on another site like http://bosun.org/slackInvite.
If you want a more advanced version with localization and error handling the outsideris/slack-invite-automation GitHub project also appears to work in gomix. See https://gomix.com/#!/project/sun-sheep for the code and https://sun-sheep.gomix.me/ for the bosun invite site.