They are all out of town. There's a couple in Christchurch which i think would be the nearest.
Couple in places like Merley, Wimborne. Quite a few in Wareham which is still pretty close , around half an hours drive or less. Beacon Hill in Poole. There's loads in the area but all a bit of a drive from Bournemouth town centre.
The Camping and Caravanning Club regularly have a Temporary Holiday Site at Parley Glade at West Parley, which is pretty much the closest you'll get into Bournemouth. Never used it and don't know what if any facilities there are but according to the listings this year the Dorset District Association have organised one there from June 9th to July 7th. You would have to become a member of the Club but as the site fee is only £8.50 per unit per night it could be worth it.
Don't forget to leave a review of the campsites you have visited this year or last!