Yes, an air ambulance chartered on your behalf can transport patients who need non-critical care, are bedridden, or fragile. Many patients take a private air ambulance just because they cannot situp or have injuries that make them fragile, uncomfortable and not suited for normal commercial aircaraft transport.
Aircraft may be equipped with a combination of an Intensive Care Unit and a Coronary Care Unit based on patient needs.
Emergency medical aircraft should also have a specially FAA approved stretcher that looks similar to a bed that allows fragile patients to travel in comfort whil minimizing movement.