Blacklands is a Guide campsite, there is a bit of woodland there but not a lot, and obviously you'd have to book in with the site. They might not like private individuals camping there - I know they take school groups though so who knows, give them a ring. Also it's not directly in the forest, it's a little way out at the other end of wierwood reservoir.
There are a couple of others - Broadstone Warren is the Scout site, it's right in the forest and fantastic - they don't take private campers though, I've asked - it's all to do with insurance apparently. They take commercial bookings if you have your own insurance. Your best bet is to join the Scouts
then you're laughing, there's loads of woodland and a couple of toilet blocks so you don't have to completely rough it (although most of the blocks are closed in the winter).
There used to be a council run yoof education site as well - Hindleap Warren, just over the road from the Scout site, I don't know if it's still there or if they take private campers - a quick Google should turn some info up.
Good luck, let us know how you get on.