As Mark says, the YHA would make an excellent base if your budget allows. The campground near Katoomba Falls would also work well.
But when you ask for a campground "in the park", are you looking for a less commercial or more remote spot? I can think of a few that are within walking distance (isn't everywhere within walking distance to a hiker?).
1. There is a small campsite with water tank and toilet in an attractive setting near the base of Ruined Castle. Four hours walk from Katoomba Station. A good base for visiting Mt Solitary.
2. There is a basic campsite at Perrys Lookdown. Probably a toilet there, but I'm not certain. Get the train to Blackheath, then two hours walk, or get a taxi since this campsite is at the end of the Hat Hill Road. Not all taxis will drive on the gravel road though. Magnificent views nearby, and access to Blue Gum Forest.
3. There's a nice little camping spot just across the creek junction below Vera Falls (at the junction with the creek from Hippocrene Falls). Train to Wentworth Falls, then three hours walk. Water in the creeks.
4. You can camp at
Upper Psyn Cave, a little way along the Narrowneck Road. No facilities. You may need to share the cave with rock climbers, who sometimes base themselves there for a week of climbing. There's often a good trickle of water nearby. No other facilities. An hour and a half walk from Katoomba Station, then off-track for a hundred metres.
5. There's another climbers' cave about 200m west of Mt Piddington, very easy to get to and less than an hour's walk from Mt Victoria Station. No facilities, but I think there is a toilet 200m away at Mt Piddington.
6. I have known people to camp at the grassy clearing where the old sewage plant used to be, in the valley below Leura Cascades Reserve and not far from the Three Sisters. Less than two hours walk from Katoomba or Leura Stations. No facilities, but there's likely to be water just upstream in Leura Falls Creek.
If any of these are of interest, I can give you more details.
Another possibility is to get yourself to Dunphy's campground at the end of the Megalong Road (south of Blackheath). It would be an expensive taxi ride, as it's a half hour drive, partly on gravel roads. But it's a lovely spot. Water, toilets, grass, views, and a good starting point for some great overnight walks. If you could get there with a couple of weeks food, you could do trips visiting Coxs River, Breakfast Creek, Splendour Rock, Kanangra Creek, Kowming River, Mt Cloudmaker, the end of Narrowneck, etc.