I suppose there could have been special "medical turbolifts", dedicated only for medical emergencies. Appropriate size, etc. for fitting a stretcher.
Was there ever an episode in TOS where we would have needed to see one?
All the (limited) canon evidence (and mostly, if not all, from the movies and other shows) suggests that the tubes won't accommodate a car larger than the one that was used. I think T'Girl
's anti-grav stretcher is a fine idea.