Hey mate - There's quite a lot of rental places right on the ring that you can rent.
I personally used RSRNurburg. All the prices are pretty similar (expensive!) but.... I mean you're there, you pay the money, you do it, and have fun, and then you can say you did it.
Note: You don't need to rent the most amazing car to have fun there. I ended up doing it with the Clio RS with paddle shifters and had a blast.
Mind you, it was in the middle of winter and -2C, so in such sketchy conditions a Prius would have been a massive handful!