You need to put the link to the editor into the registeradult and registercoppa templates.

But the problem so far is that there is no user entry in the database where the signature can be stored. So the editor will not work in these templates.

The signature editor was intended to help building and viewing the signature. You want something like controlling the signature.
This needs far more hacking and should be integrated into the vB php files.

I would suggest controlling the signature exclusive via the editor. Delete the signature input fields from all templates and just offer the link to the editor. In the above mentioned register templates you can say that the signature is definable later in the user cp.