Registration forms not submitting - What Permissions needed
Tried several tests to resolve the issue. Checked all Anonymous Roles to YES, not this problem. Turned off all WordPress Plugins, not this problem. Changed theme, BINGO!

There was a conflict with one of the JQuery/Lightbox type page overlay option within our theme that cause the forms not to submit. Disabling that option fixed the issue.

The overlay window was loading a separate page and somehow it was loading the Registration form too, doubling up the Registration form!

So, Theme conflict in WordPress.