When will happen with an Incomplete Submission when a form is submitted?

I read above that a session is kept open for about 3 months after a user starts the form, does the session close once the user clicks on the submit button?

When using Save Forms and Continue Later feature the pending/ incomplete submissions will be saved on the submissions page of your form as the Incomplete Submissions.

Once a user submits the form, the Incomplete Submission will be marked as completed and removed from the Incomplete Submissions list. You will then be able to view submitted data on the submissions page of your form. 

How to View Form Submissions

Right, but once the user clicks on submit, would they still be able to use the link to access their form or would it be closed?  I ask because we have cases where a parent would use the online application to register one child and then may go back at a later date to enrol a second child using the same email address.  I just want to make sure that using the session link won't bring up the older form entry.

No, the form will be closed or the session will be empty since their saved session data was already submitted. It will just create another session.