I am now in Japan and doing some estate settlement in Hiroshima.
The only way you can get a copy of Koseki Tohon / Shohon information is by having written and certifiable evidence that you are definitely related to a specifically named individual who may be listed in the Koseki. That can be parents, grands-parents, siblings of parents, etc.
The reason is that Japan is now very much aware of identity theft and misuse of and dissemination of private information.
Most information is already in the computers. However in a few remote districts, they still have paper documentation that much be researched. Therefore, you must be able to give a fairly detailed information regarding your relationship to your ancestral past in Japan. That will be original copies of birth certificates in the USA showing all individuals related and connected to you and the individual you want to get the Koseki for. In most cases, you must also provide certifiable identity, such as your birth certificate (original) along with passport and picture ID (original) as well as a notarized document (Affidavit) verifying your signature. That is because Japan still uses "inkan" or stamped seals.
Many in Hawaii have already used their known relatives in Japan to do a little research and get the information desired. A few are using paid researchers and companies to do the checking and get you detailed book bound history, costing anywhere from $2,000 to $10,000. You might find that hiring such a research person/company is much thorough and well documented, easier and in the final analysis much cheaper to get the results you want.