Of course. But it should be more than a simple mission given to him by the Gotei 13. I don't know if Vengeance had any deals that went bad. But, I was planning on having Mikado hunt him because of a "job" that personally cost him.
She's a target too. I wanted Mikado's history to be tied with the Coven, somehow. So, I figured that they would have a hit on Mikado's parents after he was born to give him proper motivation to seek them out.
Should we pick it up on present day? Or, during a time period parallel to that of Turn Back the Pendulum where Mikado would be as old as Byakuya but training under Nanashi? That's where he'll finally be ready to ask his grandfather about the fate of his parents.
He's stern and benevolent, yet not exactly willing to answer questions about Mikado's parents. If you want, Naomi could be the one who recruits Mikado for a mission from the Gotei 13, that would divert the conversation while simultaneously introducing Vengeance into the present.