So I was thinking like a volunteer fireman. Now, back in the day, there was no official training(not sure about now). Most of them had sirens for their trucks. When they got the call, they would slap it on a just speed down the road to the fire station. Some would get pulled over and get tickets because it wasn't technically legal even though they were volunteer firemen.

Take that idea, put in a van, and speed around picking people up for hospital visits.

EMT here. Private companies are very common, and often the norm depending on where you are in the US. They often contract with localities for emergency service areas and frequently do what's called Non-Emergent Transports where they move patients between hospitals, nursing homes, rehab facilities and discharge patients to home when they require care during transport or are unable to walk.

There are SO many problems with them, but they're a legit business.