Permission to submit webform only for the author of the node
You haven't said if this is Drupal 7 or Drupal 8 but this should work for D8

It sounds like each author already has a user account on your website, so you can go to "Settings" on your webform and then click the "Access" tab (.e.g /admin/structure/webform/manage/FORM_MACHINE_NAME/access)

And in there you can restrict the ability to create submissions to a specific user(s) only