When I worked for a private ambulance company our policy was to either give it to PD or, if no PD were on scene, stuff it in a lock box we had on the ambulance.
We would let the hospital know we had a firearm onboard and their security would come grab it from us and store it for the patient.
I would almost always try to go with option two as I'm sure it involved MUCH less red tape for the patient to get their firearm back.
edit: A partner of mine went behind my back and reported me to one of our supervisors because I unloaded a Glock 19 a patient was carrying before storing it. According to her unloading it before locking it away was "Dangerous" and I shouldn't have handled the gun any more than necessary.
End of the day even if that specific company in the city I worked in were the ones to pick you up there is no telling what would happen to your gun. Ask 5 EMT's from there what protocol is and you'll get 2 different answers and 3 blank stares. We had an EMT get written up because he found a knife on an unconscious patient and decided the best course of action was to toss it out the side door onto the side of the road and leave it....and then document that he had done it in the report.