Check the nearest large city. For example, in Pennsylvania anyone in the state can sign up for the Philadelphia library even if you've never been to Philly. The process is entirely online.

I've also been told some libraries will let anyone sign up online for a yearly fee. Then you use the Libby app to borrow books.

Just call them. I'm sure the librarians would be happy to help you out, especially since many libraries offer ebooks.

Pull a reverse gattica, and hire someone else to get a library card and assume their library identity.