Civicrm Custom Profile Wordpress
Not as a built-in CiviCRM feature.

Perhaps you don't want to use profiles for this, but could instead use a Wordpress specific form builder? You need some integration with Wordpress because using a state related to whether or not they've filled in a form would be handled within the CMS (which handles user authentications, sessions, etc.).

I believe Wordpress provides a couple of form builder options that integrate with CiviCRM.
https://docs.civicrm.org/sysadmin/en/latest/integration/wordpress/forms/

Alternatively, you could replace the permissioning with workflow (in a limited way) and use the profile but only link to it at specific times, e.g. via the membership thank you message.