This is a pretty hard question to answer on SDN because it's going to vary by state and probably by local jurisdiction. In Oklahoma, you would have to have a Class A Chauffer's license. Individual EMS services may have varying requirements based on what their Commercial Liability insurance policy requires. Not very helpful info, is it?
Call or check the website of your local ambulance company. This varies from state to state. In CA, for example, you need an ambulance driver license in addition to your EMT cert. This is also something that they should cover in your EMT class. In any case, unless you have a DUI on your record, getting the extra license should not be a big deal.
I worked for a private ambulance company in Chicago and while I never drove, I had to change my license from out-of-state to Illinois so that I could be covered by their insurance in case I ever had to drive. I just had a regular non-commercial license so I'm pretty sure that's all you need. I don't know about Cook County city requirements though...
I worked for First Care Ambulance in Aurora, and we just had to give them a copy of our driving record and have no more than two tickets I'm pretty sure. If you're living in Chicago you'll probly be working for Superior, so give them a call and see what their requirements are.