Taiwan is not a party to the Hague Service Convention and has no service treaty with the United States, so there is no Central Authority route. U.S. litigants generally choose between Letters Rogatory, which are transmitted through the American Institute in Taiwan (AIT) because the U.S. has no embassy there, and informal methods such as delivery by a local agent or by mail, where the court's rules allow them.
Letters Rogatory to Taiwan are formal and slow, and they must be prepared with care: for example, they should not refer to Taiwan as the "Republic of China". Informal service is much faster; U.S. courts have accepted methods such as international courier delivery under FRCP Rule 4(f)(2) where Taiwanese law doesn't prohibit them. If you may need to enforce a judgment in Taiwan, take advice from Taiwanese counsel before choosing, because Taiwanese courts look closely at how the defendant was served.