What is the best way for the person who submitted the form to edit the form?

Our example would be, a shipping request form.

Once the form is filled out, can the person make edits to that form after it has been submitted.  For example, they want to add items to ship or edit part numbers

Yes, it is possible to let your users update their submissions data anytime they want by providing them the edit URL.

Here's how: http://www.jotform.com/help/40-How-to-Let-Users-Update-Their-Form-Submissions-at-a-Later-Date

Thanks for the answer, there is no way to do this without sending them the edit url?

Yes, there's no other way other than sending the edit URL. The edit URL retrieves the data and populate it in the form. They can update the data and resubmit the form to update their submissions.