"What transportation can we take in order to get to the hospital?"
Is this sentence strictly ungrammatical or just unnatural to say: 我们可以坐什么去医院?
Is this related sentence ungrammatical as well “我们可以坐地铁去医院” ?
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).
For me it sounds alright.
But it would be even better by changing from a What question to a Which question like this:
As to the response, it would be more natural to omit
Firstly if your question asks 什么/Which then you can't attach 吗, which is for Yes-or-No questions only.
Secondly the term 运输 is usually used on transportation of goods. For passenger transportation, we can simply say 车 or 交通工具 or just imply it like in Duo's sentence.