I think this would be interesting to have. There were times as the Vigilante where I would've liked to write a death note as to why I killed them, or a wisecrack or something. Maybe not like the normal death note but something similar to the Jailor's version of it.
Based on a wiki, all Town Killing roles used to have a death note, but was removed as "They used this to easily confirm themselves, which is the reason BlankMediaGames made the change.". But usually, they can be confirmed easily or are rarely challenged, and they usually claim on the next day for doing something. In a way, the Vigilante and the Jailor are quite similar: they specifically attack one target if they want, they have a limited number of attacks, and they take a penalty for wrongful killing. So why not let the Vigilante have one as well? Curious to see what people think.