This was never revealed. We will not know until the author chooses to reveal it. But if you think about it one of the conditions of being enabled to be a captain is to have achieved bankai.
it has never even been stated that the Captain of the Kido Corp has to have a Zanpakutō, when Tessai Tsukabishi was captain of the kido corp he was seen holding a staff, however if that staff was a Zanpakutō is unknown.
Probably being a Shinigami, everyone from Kido Corps must have Zanpakutō, but in what shape, is still unknown. As we can know, Hacchigen Ushoda, previous vice-captain of Kido Corps has got a Zanpakutō, so Tessai so far must have.