The public library system has 6 branches on Kauai and 2 public librararies on the island are open on Saturdays (Hanapepe and Princeville). If you find books that are located in other branches across the State, you can order them through their interlibrary loan system. You need a public library card and a PIN number to order.
If you want to search the biggest library catalog, try WorldCat, which allows you to search the collections of most libraries in the United States and several thousand libraries around the world. WorldCat contains catalog records of over 1.5 billion items.