Probably talk to the Sister city org. You may find it hard to get your entire family with another family though.

There is an organization, Hippo Family Club (now called LLP), who do homestays. http://www.lexlrf.org/hippo/index.html One night billeting for a full family… not sure if they do that, but at least you could probably see if one of the Hippo families would volunteer to take you for the night.

Do you have any Japanese friends, or friends of friends? That is the best way to get a homestay!

Good luck!