As far as I know, there's no such registry. There are exceptions under Section 108 for institutions that fit a certain definition of library, but from what I can tell as a non-laywer, they don't allow the kind of indiscriminate reproduction that the IA engages on: https://www.law.cornell.edu/uscode/text/17/108
From Wikimopedia: "The Archive is a member of the International Internet Preservation Consortium and was officially designated as a library by the state of California in 2007."
