No, not that I know of, although I no longer drive ambulances in Nevada so don't take that as gospel. I am a firefighter and we do need a special endorsement on our license to drive fire trucks. The main thing about the endorsement is that we have taken the test for using air brakes. Obviously the truck is a lot bigger also. So, I am still driving "ambulances", but I have the "F" endorsement, so I don't know what someone working for a private company needs anymore. FWIW: I worked for a private ambulance company in Las Vegas for 7 years prior to getting on the FD and didn't need any kind of special license then. But that was 16 years ago now.
If it is important, I can easily find out if you need anything special. I know for a fact that the local private ambulance companies have had insulin dependent diabetics working for them FWIW.
I have worked as a paramedic in Ohio, West Virginia, Pennsylvania, California (all pretty briefly), and Nevada.
The only place you needed an ambulance driver's license when I was working in those places was California. When I was working in California, it would have been about 1986 (I am getting really old). At that time, you could be an ambulance driver and not even be certified as an EMT. You would just drive and your partner would handle all the patient care. And, this was actually done frequently. There are tons of private ambulance services in the LA area that don't even want to run 911 calls: they do interfacility transports of patients that actually have insurnace (usually medicare). So, they arn't doing much emergency medicine: just driving people around to dialysis appointments and what have you.
Again, that was a long time ago, and things have probably changed since then.