You guys are forgetting something. Leon and Aerith already explained to the gang what happens when a world is consumed by the Heartless: it disappears, as does every living creature on it. That doesn't mean they become Heartless themselves, so they don't have Nobodies either. That's why Selphie is perfectly fine in KH2, along with everyone else on Destiny Islands. Once Sora and Mickey sealed the door to Kingdom Hearts (and defeated Xehanort's Heartless) the worlds were restored and everything went back to normal. Kairi returned, but Sora and Riku were missing as far as the islanders were concerned.
A few days later, Sora arrived at Castle Oblivion and slowly began to lose his memory. We saw at the beginning of KH2 that Selphie no longer remembered him, she thought Kairi was making him up. So his parents obviously don't remember him either. According to an interview with Normura, they all knew that "on the night of the big storm, Riku disappeared". I imagine in the beginning it created a frenzy and Riku's parents were worried, but as time went on and they couldn't find him there wasn't too much they could do. Then during KH2, they all remembered Sora, but it was pretty much the same thing with Riku: what more could they do when he obviously wasn't there? We don't know how they reacted to Kairi's sudden disappearance. It would depend on whether or not anyone saw her with Axel on the beach, and how long she was in the World That Never Was.
We actually do know some things about their families. We know Sora and Riku's parents are alive, because the latter mentioned them just before he was sucked into the darkness. "Once we step through, we might not be able to come back. We may never see our parents again. But this could be our only chance. We can't let fear stop us. I'm not afraid of the darkness!" or something like that. So they both had living parents. Kairi had a grandmother in Radiant Garden obviously, but her parents probably perished when that world was destroyed. She was sent alone to DI, which we learned from the flashbacks Sora had; she was staying at the mayor's house when she first arrived. Either he took her in or found another couple to raise her.
It IS a bit concerning that Sora didn't seem too worried about his own family, but I'm sure he was to some degree. To be fair, the last people he saw on Destiny Islands were Riku and Kairi, right before they were each swallowed by darkness. Naturally he's going to be more concerned about them, especially after he hears from Leon and Aerith that the rest of the Islands disappeared. That would mean there was nothing he could do for his parents. Might as well concentrate on the people he knows he can save. Riku was obsessed with leaving in the first place and was slowly being taken over by darkness, so I doubt he was very worried about his family. Again, he knew he had a chance to save Kairi (and at one point, Sora) so they were his main concern. Kairi was in a coma for most of the game, and afterwards she knew that Sora pretty much had everything under control.
And all this info is practically pointless because like everyone else already stated, the families of the heroes have little impact on the game. But it's still kind of interesting to know their backstory.