Yes, there is such a thing as a stingless bee that produces honey. They don't produce a lot of honey, but will produce some surplus to harvest under the right conditions. I know some are in Austrailia (See http://www.zeta.org.au/~anbrc/honeyproduction.html ) I don't know of any in the US and importing them probably isn't an option. Note though, that they do have defenses. This will bite and apparently have chemicals in their saliva that can cause pain. May not be as bad as a sting, but they can bit unlimited times.
With proper protection, a smoker, working methods and a gentle breed of bees such as Carnolians, stinging should only be a minor issue.