It might be possible to do this via a View that lists people that are linked to the contact person's Org and provides customized links from the view to a webform (passing contact IDs via URL to populate the webform). You may also need the [Relationship Permissions ACL extension](https://civicrm.org/extensions/relationship-permissions-acls) and have these relationships set up to grant editing permissions.

e.g. If you have a view that is pulling the current user (contact person with permissions to add/edit), you can use the View to also get the Org and the other members based on relationships, then feed them into a Webform with CiviCRM functionality enabled, feeding the contact IDs into the webform using the URL. ("If the url contains e.g. cid1=123, it will be used to pre-populate this contact (takes precidence over other default values).") 

You could set up the form to only allow input for 6 contacts (the org and 5 members) and specify their relationships in the webform/civicrm settings, you can feed the contact IDs if they exist, or allow new contacts to be created if they don't.