Generally, Japanese has strong traditional culture called UCHI (internal, close family/friends circle) and SOTO (outside, not close in terms of relationship). So, we don't have such culture to accept strangers nor just acquaintance even they are Japanese.
If you want to stay at some home as real japanese, you can try " Airnb" with share apartment /house.
Airbnb is getting popular but there are still obstacles with Japanese laws. The government is trying to change the law to allow services like Airbnb due to the increasing number of visitors to Japan and lack of hotel rooms especially in the central Tokyo and Osaka areas.
Unless you personally know someone, or are on an organized trip, then staying at an average Japanese person's home is not going to happen.