"What transportation can we take in order to get to the hospital?"
To my (Taiwan-trained) ears 我们可以坐什么去医院 sounds a little odd. I would say 我们可以怎么去医院 (or even better: 我们可以怎么到医院). I suspect that one big factor which makes 坐什么 sound weird to me, is the fact that it can cause confusion because it sounds exactly like 做什么 "do what" (that confusion can be cleared up by saying 我们可以坐什么交通工去医院 "what means of transportation...").
我们可以坐地铁去医院 is perfectly fine (Taiwanese people use a different word for "subway" (捷运 jiéyùn) but that's a minor regional vocabulary difference and 地铁 is definitely understood in Taiwan as well).