Ok, in-game, a vampire is made when a vampire bites a human female but doesn't drain enough blood to kill her. With that established, it's implied that Lynette Kojima met Kana and Kagome, most likely at one of Lynette's book signings. This was during the interval between MC being their baby sitter and their later reunion with him at the cafe. Kana most likely bonded with Lynette over a mutual love for writing, with Kagome tagging along to not be left out.
Another in-game rule is that, to protect vampire society (and Myth society in general), newly created vampires have to leave their old families and move in with the clan of the vampire that made them.
After the twins' initial meeting with Lynette, it can be logically deduced that Lynette became intimate enough with them to bite and turn them. Lynette, being the First Generation of the Kojima Clan, had some discretionary leeway and allowed the twins to decide if they wanted to be with their biological family or her. They chose to stay with Lynette.
We might discover more on this as the game progresses.