I've seen mommies use hot tubs and if we applied the same concept to the onsen (japanese bath), I reckon you can use it but make sure that the usual hygiene and safety precautions are observed.
Be sure that it is clean and well maintained, as organisms found in such places can cause a wide variety of ailments, including skin rashes and infections, urinary tract infections, irritation of the digestive system etc. Before breastfeeding, make sure you take a thorough shower. Also, if the temperature of the onsen is too hot and you feel discomfort, best to step out and take a breather first.