I'm not sure exactly what products you are referring to, but since you mentioned wheel cleaners, I would NOT put any wheel cleaners which contain HF or ABF into a foam gun. If you use other acid or caustic wheel cleaners, please be very careful about incidental splashing of these onto you, shrubbery, or other non-wheel surfaces of your car or house.
Also note that wheel cleaners are usually not meant to be diluted (except in the case of some of the HF/ABF types), so that may reduce their cleaning power, and that use of more corrosive cleaners my accelerate corrosion of the brass parts of the foam gun, which means you should follow the directions and flush with clean water after use (which I never do for regular car shampoo).
On the other hand, the foam gun's main market is for industrial/food preparation cleaning and sanitizing, so people certainly use it with some strong stuff, although germ-killing chemicals are certainly different than brake dust-eating chemicals.