How to limit webform submissions access to only users who have Webform submissions?
Maybe not a complete solution but here is how you can start of


create a view that displays submissions 
make it available to everyone (view has permissions) 
put it in a URL that is not linked from your main site (its like being hidden but not exactly)
put the URL link at the submission completion page (thank you page)


beware this will not prevent a person that knows this URL to visit it and see the results.  I am not aware at this point of a module that will give the required permission see results after submission or something similar so if the above is not to your satisfaction then you might want to create a module to do that or modify webform to include such a feature