There is no online system for getting koseki. Jurisdiction over koseki lies in the local municipal government and not the national government, and these certificates are still paper-based, because the old koseki-touhons are written in kyu-kanji, many of them now obsolete but cannot be converted into modern kanji because koseki is legal documentation that cannot be tampered with.
The only way to get the koseki-touhon (or shouhon, if restricted to the individual in question) is to write or phone the appropriate municipal government, explain the situation and receive instructions on how to go about it. As long as the boy can present proof of identity, the municipal government is sure to send him a certified copy by mail.