I am assuming Budd was involved with the Taxi only because they were stamping '28,'29 Tudor panels ,also the Deluxe Delivery [ as shown by Vince Falter/Forever4's photos on previous posts] as well as other body stampings for Ford.
I have never read they were done by Yellow or Checker ; they were a variation of the Tudor body by the looks of it.
I have just checked Vince's excellent site, & it appears the Taxi Cab body was built by Briggs, so I am wrong in my assumption it was a Budd body.