Not camped there ,but Swanage is a lovely little seaside town ,weymouth is a lot bigger and more commercial,pretty sure most of the camps will be fine as long as they list all facilities should be fine .....tank museum opposite is worth a look too....my fav site would be nearest corfe castle and swanage but the reviews of the showers are not great ...... campsite review
Loads around that area, Lulworth, Corfe, Or even Rockley park in Poole, not far.
Rockley is the only one i have ever camped at, and that was a long time ago.
Absolutely love that neck of the woods - anywhere from Weymouth across to Bournemouth - every year we spend our main holiday in the 'van at Black Knowle Caravan club site in the New Forest - so if you fancy 'stretching to that far there is a massive camp site right opposite this site and its like travelling into another world and would whole heartidly recommend staying in the New Forest even if it is a bit of a stretch from Monkey World. Just like the other guys i can recommend the tank museum, Corfe Castle, catching the ferry across Poole Harbour (the nudust beach at Studland emmmmm). Bournemouth, Weymouth (as already mentioned), in fact I could go on and on. Make a long stay of it and I guarantee you will be going back year after year