We love Blacklands Farm in Albourne. Friendly site, lots of kids, reasonable facilities and you can hire a fire pit which is lovely. Reasonable too and you can pitch up around the edge of the field with a good view of the play area in the centre.
Well there are quite a few campfire type campsites in that area, particualrly round East Grinstead way - St Ives Farm, Chafford Park campsite, for example. Wowo I think is a bit further towards or in Sussex itself. Even Forgewood is on the Kent/Surrey borders, as is Manor Court Farm. All lovely campsites, ideal for children, and all allow fires.
It is also worth having a look on www.ukcampsite.co.uk and using their map search facility so you can see exactly where the different campsites are.