At the risk of being nitpicky, it's actually "kazoku-buro", literally "family bath". Many onsen ryokan, ie places that do offer accomodation, have them, and it's usually a large marketing point for them. I haven't personally stayed at a place that features them (although I have stayed in places that have rotenburo, or outdoor baths, attached to each room--agreed that these are more romantic than family-friendly. for one thing, they can be quite small!). When I've looked at places that offer kazoku-buro, it seems you usually make the reservation to use them when you check in, for a block of time of around an hour or so. Some places charge extra, some don't.

If you don't read Japanese, it might be easiest to go through a Japanese travel agent who could point you to places in your desired area and price range. I imagine a place like JTB (Japan Travel Bureau) could be helpful.