Classfull registration in RAdb.

I am currently registering routes for the company I work for and
I am wondering whether I should register customers classfull space.

I think that you should; if you are the origin AS, and people can't
find a route object for a route that you inject with yourself as the
origin AS, the question of why the route object hasn't been created is
more likely to go to you than to the owner of the address space.

Additionally, you have all the data necessary to do a complete job. If
you delegate the task to your customers, some will do it, some won't.

If the address space is portable, but they do not have an ASN, who
has the authority to regsiter the route object to a particular ASN?

The owner of the ASN used as the origin AS should have the authority
to create route objects to document previously-undocumented address
space, in the interest of a better/accurate RAdb.