Well, Kudo Yusaku was both involved in the Kid case and close with your father; I think it's highly likely he knew your father was Kid, and probably at least knew something was suspicious about his death ... and I'm not sure on the circumstances, but Kudo Shinichi has obviously gone into hiding badly - as if he's avoiding someone where he expects any clear confirmation of his status and location would be lethal. I thought it was unusual; it would have to be some kind of organized group larger than one or two people to justify being that paranoid, and wiping himself off the face of the Earth would be a bit extreme for most normal mafia groups, so I had been wondering.
Conan mentioned that Shinichi had said that one of Toichi's apprentices bothered him - he said he didn't know any more than that, and I thought it was odd, considering he would've rather young at the time; it occurred to me that if one of your father's apprentices had seemed suspicious, it probably wasn't a coincidence. If Shinichi knows about Vermouth, then he knows about them, which would mean that they're the people he's hiding from.