Here's a few to start. Keep in mind that I'm more of a backpacker than luxury safari kind of gal so most of the below aren't fancy, but they're all friendly and welcoming to self drivers.
Livingstone - The Waterfront has about a dozen parking spots for self drivers, each equipped with a sink and firepit. Nice showers and toilet blocks. You can also book fixed tents or rooms. On site restaurant, bar, two swimming pools and activities booking center. Located about half way between town and the falls, so neither are within walking distance, but its right on the river and gorgeous. Extremely helpful staff. Vervet monkeys will steal your laundry and jump on your tent for free.
Chipata - Mama Rulas has a grassy area for tenting, rooms available as well. Swimming pool, bar. Convenient location on the way in or out of South Luangwa.
Lusaka - Eureka Camping Park has a huge grassy area with covered spots for tenting ($7/night). Small freezing cold swimming pool, cozy bar and a few rooms for rent. Bonus animals running around include semi-tame zebras and impala.
Luangwa - Bridge Camp is pretty much the only option along the long road between Lusaka and Chipata. It's fairly small and can only fit a few vehicles ($8/night), but they have a few cabins as well. Pool, bar and activities like birding and trekking.
South Luangwa NP - So many options here. I've stayed several times at Croc Valley. Lots of grassy areas for tents and room for a few 4x4s right by the river ($12/night). Cabins available also. Pool, restaurant, nice bar. They do game drives and walking safaris in the park for reasonable rates. Elephants wander through camp and expect to be harassed by baboons and vervets. Good value.
Chirundu - Zambezi Breezers is a small campsite and lodge right on the river. Tiny swimming pool, restaurant and bar. Huge grassy area for tents and lots of space for 4x4s ($10/night) - hippos will come visit at night. Offers fishing trips for tigers and can arrange canoe safaris into Mana Pools area. Super helpful staff.