This map is from worldatlas.com, which offers detailed maps and information for each Asian country. Some countries on this map are also on the map of Middle East countries.
RLG's East Asian StudiesCommunity offers a Guide to Human Resources in the East Asian Community, which identifies collections that are strong in specific areas as well as institutions with especially knowledgable staff.
The East Asian Libraries Cooperative World Wide Web identifies its goal as "to support research on East Asia by creating electronic resources and developing web-based information services through cooperation among librarians, scholars, and researchers, and by mirroring resources developed elsewhere to facilitate access." Although it does not discuss document delivery as a means of access, it may be helpful.
The Council on East Asian Libraries includes much information on doing research in Asian Countries. The site includes a page on East Asian Library Websites iin North America.
I've broken out some of the pages under the specific countries below.The European Association of Sinological Libraries maintains a list of libraries with strong Asian collections, including substantial information about the library in a database form.
China: The Council on East Asian Libraries site includes a page with links to libraries with Chinese collections.
University of Virginia's list of links to library catalogs in China, Hong Kong and Taiwan
Japan: North American Coordinating Counicl on Japanese Library Resources Global ILL Framework. You will want to print out their PDF guidelines
The Council on East Asian Libraries siate includes Digital Reources Guide for Japanese Studies as well as a list of Japanese Newspapers in the Harvard -Yenching Library.
Korea Collections in North America with links to catalogs and contacts.
To find listings of libraries for a specific country, use the links from the Repositories World Wide page.
