Where do I start moving a person in a coma from Cleveland, Ohio hospital to an Illinois hospital?

Contact the Social Services Department if the person is at one of the hospitals here or the Social Worker if at a facility. They can help you arrange transportation, work with the doctors and find a facility in Illinois.

If Medicaid is involved, it doesn't go over state lines.

Well the government won’t pay for it is my opinion. Not under normal circumstances. It will be private pay with an EMS service I’m guessing.
It’s a little over five hours from Cleveland to Chicago. You don’t say where in Illinois you need to get the parent but regardless the distance is probably not the deciding factor.
So it’s drivable but what medical requirements does a person in a coma need while in route?
Then you have to have a hospital willing to take the parent.
Different states have different rules for Medicaid. So there is that. Is this person a citizen of Ohio or Illinois? That might make a difference.
My inlaws got evacuated because of a hurricane 7 hrs from home. We finally got FEMA to say they would bring them home. FIL died the day they were to be transported. MIL made it fine although she died a month and a day later. FIL was brought home by a funeral home transport without a problem.
Im sorry for your difficulty.
Hopefully someone will know how to manage this.

Medicare won’t pay to transport patient to be closer to family. Advantage plans won’t either.