From what I've observed there's nowhere near the amount of sponsorship in the sport in America - even aside from the BBC not allowing visual advertising, it just doesn't seem to be in the culture here. When sponsorship does happen it tends to be in the form of free parts and free services on a local level rather than anything formal with logos on a big banner in the pits. Battlebots have gone in a slightly bizarre F1/NASCAR sort of direction in that regard - everyone had those big banners in the pits with a space for sponsors, and I wonder what would happen if you'd entered with none...

There are exceptions - the old Team Storm sponsors page is still up and shows that it is possible to get quite a lot, if you're Ed Hoppitt at least - but generally I wouldn't count on it.