"What transportation can we use to go to the hospital?"
Is 我们可以坐什么车去医院？not correct? Unclear to me how 什么去 works without 车, though in context it makes sense.
I think of it as "We can take what to go to the hospital" in Chinglesh, the 车 would mean car in the context of your sentence so wouldn't be correct. The word transport is not present in the actual sentence though, which is slightly tricky.