We currently do not support saving the form state or modifying a submission a user has already submitted.
This should be an idea on our ideas page where you can add or vote on feature ideas:
If you need to add a new idea click "Create an idea" under "Actions" in the top right.
For the case of the PDF embedded in a site, if the user downloads the PDF then they could save the state of the PDF, but could not modify a submission they already made. Let me know if you have any more questions around this.