I’m not familiar with Sendy, or with how it is integrated with your WordPress site, but there seems to be Zapier support for Sendy that can be triggered when a subscriber is added to a mailing list: Sendy - Integration Help & Support | Zapier.
If you can get WordPress/Sendy/Zapier integration working, you can automate invites by having Zapier create a POST request to https://forum.example.com/invites when a user is added to the list. An example curl request for creating an invite looks like this, with $api_key set to your All Users API Key. Set the api_username to the username of an admin on your site:
curl -X POST -d "api_key=$api_key&api_username=simon&email=inviteduser@example.com&group_names=your_group_name" https://forum.example.com/invites
You can test this out with curl or Postman before trying to set it up with Zapier.