However, when the public attempts to submit a ticket they cannot enter anything in the text area part of the form.

In Editor Profiles (which you can access from the JCE Control Panel), you may have restricted the "Default" editor profile to Super Users only.

I would suggest creating a copy of this profile, assign it to the usergroup who will be accessing it (if this is a non-registered guest user, then this would be Public or Guest) in the Setup tab, then make other adjustments to the editor profile as required to restrict features for these users.