I know RingTaxi are long established offering rental and accompanied driving on the Nürburgring Touristenfahrten.
Touristfahrten is what Mark refers to. The Nürburgring Touristenfahrten is essentially a German public toll road with no speed limits. Your driving experience would be within one of the managed Touristenfahrten sessions which run daily in season. You'd be mixing with anyone and everyone basically. The GP track is something different, you'd have to be competing to drive on that in most instances.
As far as I know there's several companies doing a similar thing to RingTaxi, based in Nürburg. I don't know enough to compare them, but I know RingTaxi has been around for a long time. Getspeed Racetaxi and RSR Nurburg are another couple Google points me to.
It's a shame Apex Nurburg no longer offer taxis and rental. They ran that business for a few years, but are now focussed on car prep for racing and such things I gather. Run by a Californian called Robert and they have a neat little place in Nürburg. They used to offer a couple of hotel rooms. I saw a lot of them through a vlogger called Misha Charoudin, who used to work for them.
edit: The former Apex Nürburg premises are now owned by Nurblife, doing a similar sort of thing, maybe at the lower end of the scale.