There are a lot of well to do people around there and no obvious places I can see where you can wilc camp. There may be a couple of places back towards South Harting/Petersfield just off the South Downs Way where you can camp up.
There is a car park up on the Downs overlooking South Harting but I don't know how the are about overnight camping. I know Goodwood has a camping field for the Festival of Speed but as you're there at the Aerodrome they may let you camp/park up on the estate.
You don't ask, you don't get is my motto, give them a ring and see what they say, there's plenty of parking areas over on the track/aerodrome side, I'm sure they'd let you sit up there.
Failing that, the Witterings aren't a million miles away, Scotts Farm at East Wittering.
Do you mean the car park round the corner from the horsy circuit? That was one that I thought about but have heard it's a "Dogging" location! lol! (That may have a created a little mor einterest in me lol) A