Note that programs that state they don't sponsor visas might not accept J visas either. In fact, I would assume that's the case unless they tell you otherwise.
When they say "don't sponsor visas", I would assume that includes both H1/J1 visas; otherwise they normally distinguish and say they only do J1 (of course, most of the processing for J1 is done by ECFMG as opposed to H1 where the program has to go through complicated processing)