I haven't tried this myself. I've only run Applets and gone through the hoop of asking permission for saves & loads. However, I suspect you don't want to be dealing with the PersistenceService. You just want to save/load on the client, without having to deal with permissions, yes?
Did you see this page? It compares Applets and WebStarts. http://docs.oracle.com/javase/tutorial/deployment/_riaDecisionGuide.html
If I read it correctly, a WebStart should allow you access to the client file system, but will have limited internet access.
If you want to have both
web access and client access, I don't know that the client access part is possible without using signing or at least without asking permissions, or via bumping up a level in technology to run a multi-client server (which I also have not done yet).
Did you see the example program called "Notepad" in the WebStart demos? I ran it and it has a File/Save and File/Load in the menu bar, but these haven't been implemented. In my mind, their existence in the menu bar suggests these capabilities are possible but were just not included. Is the source code posted? (There is a code examples area in the tutorial.) Maybe there are comments in the code which say something about how these would be implemented. Or, try plugging in the standard Swing file service to their code and see if it runs.
Or maybe you will get a reply from someone who has actually done this and knows the answer!