You can if you want Sign up at the Jackie Stewart pavilion (located just before the track) for a donation (usually ranges from £10 per person) and sit in the passenger seat for 7-8 fast laps, from a Ford Focus RS to a McLaren 570/650 or Ferrari 458/480.
These are the drivers own cars so I cannot predict what may be their but it's usually a good range. If you want to get a ride you must get there early and speak to the drivers before getting wristbands because it's quite difficult to get a space. If you know any drivers going thats good as they will try and get you in. On the 1st day I recommend speaking to the drivers, then the 2nd event that occurs try and say if you have a space and they will get you in before 12 hopefully as thats when it's their lunch break and everything gets boring. You can, once you have the wristband, go round a further 7 laps if they haven't got any busy slots. I'd say get to know at least two drivers so you have a backup or an extra ride!
Disclaimer(?) : the event to attend is free and watch and photograph cars (just go into goodwood grounds and park in the paddock) but to actually take part you may pay a donation. (gratefully) along side this, Peter Saywell (The organiser) brings down his spectacular array of cars which sounds a bit like this; A La Ferrari, Ferrari TDF, Lamborghini Aventador S, Porsche 918 AND... A Pagani Huayra BC (as you may have heard on the A27 A Pagani BC crashed, that was his, he managed to restore it)
At one point he had two Paganis. Just another reason to go. These are stationary but do go out on a parade lap during the lunch break with the other cars, so you can hear the pagani, la ferrari, 918 (unless in hybrid mode) and aventador S.