I'm a little confused about your requirements.
If you don't need voice calling, then the many data sims out there are the best value for tourists, which don't need a visa.
If you need to set up a contract, then yes, you will need identification and a Japanese address to send bills to, depending on provider. Some also do credit checks too if there is any discount handset involved.
If you are visiting from a country with a network provider with a roaming partner, check that out as you may find that it is included in your monthly fee for certain networks without any extra charge to you.
For Voice SIM options, you'd be looking at rentals, e.g.
Hope that helps!