I'm not 100% sure yet, but the 'From processor' direction looks very promising to explore. https://civicrm.org/blog/jaapjansma/using-civicrm-form-processor-extension-to-handle-form-submissions-from-an-external.
At least it can open a CiviCase for a user who has just entered his name and email.
I don't think you can use civi's profiles with cases this way, partly because you'd want to collect some contact data as well and you can't mix in a profile.
In drupal I would say use webform_civicrm. For wordpress I believe caldera/ninja forms can do cases but there was some talk about it being discontinued at some point.