Can someone edit their submission or do they have to submit a new form?

If so, how do they do it?

If you are referring to the form owner to edit their submission, then you can do that from the form submission list.

Please check the following walk through image to get the idea:

If you are referring to the user who submit the form, then you can add an edit link to the autoresponder email.

Please follow this link to know how to do this:

https://www.jotform.com/help/40-How-to-Let-Users-Update-Their-Form-Submissions-at-a-Later-Date

If you have any further query, please let us know.

Great!  The user is who I am referring to.  Many thanks!