Yes it is possible to rent a car and have a tourist drive. There are two tracks that you can try: GP track and Nordschleife. If you want to be safe, then you should try only the GP track. The other is the big one and it is quite dangerous, especially on open (tourist) days. They call it green hell for a reason ...
Plan your trip before you go and check the availability dates in
https://www.greenhelldriving.nuerburgring.de/#/landing