Connections does not have user registration. You can use the core WordPress registration feature or a plugin that extends it. There are many freely available plugin that cover this.
As far as submitting to the directory itself, yes, that is possible.
If you do not intend to permit the user to maintain the entry after submission then all you need is the Form extension:
If you do want the user to be able to maintain their submission, then you’ll need the Link extension:
Form becomes optional, but recommended, especially if you want to allow maintenance on the frontend vs their WP admin user profile page.
Hope this helps! Let me know.