Is it possible to move a vent from the floor to the wall without ducts?
A vent isn't hard to extend.  Any decent contractor could extend your current vent to closet wall.  I am assuming this is the outside of the closet too - you don't ever put a vent in the inside of a closet and not even sure that is code.  Anyway you need to part ways with this contractor if you can. If he even thinks about telling you something that dumb then who knows what he is doing when you aren't watching.