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Yes, as explained above, you should to make sure that you have granted the necessary permissions to your sub-user account, so they should be able to edit and view the submissions.
This guide will help you to set properly your sub-user account: http://www.jotform.com/help/232-How-to-Share-Forms-with-a-Sub-Account-User.
But, if the issue is still there, you may share us the username of your sub-user, we will take a look on it.
Noone can access our forms – not even our clients on the published link which they have always managed to do.
I did check your account and found that you have shared your forms with only one user with read access. Please check the screenshot below:
That is the reason why your sub user is not able to edit forms. You will have grant proper access permission to your users and the following guide should help you: http://www.jotform.com/help/232-How-to-Share-Forms-with-a-Sub-Account-User
There is a little confusion when you say "not even our clients on the published link". I am not sure if you are talking about edit links. Please share one of the link and we will take a look.
We will wait for your response.