G'day
In all the public campsites we went to there were showers and toilets, with separate male / female blocks. Some showers (a few) had hot/warm water, most had only cold water with varying pressure. We found that washing Indian style worked best; a plastic bucket and ladle (i.e. fill up bucket with water, soap up, wash off with ladle) simply to overcome the pressure issues, esp when there's many people. You can get a plastic bucket on the road.
Toilet paper is "sort of" provided at campsites - not much at all and usually only when the toilets are cleaned in the morning. If you are going in via Arusha or any city stop at a store and buy a supply; there's plenty around (contrary to some reports) so don't waste the space in your baggage!
I'm assuming you're going with Lion King. They have powerboards in the car that work, so if you take adapters your devices / cameras will always be fully charged. At campsites (as is usually the case) there are some powerpoints but demand/competition is high.
Food, drinks and water through Lion King are totally adequate; their meals are huge; the campsites don't have stores or kiosks to get anything but honestly, you don't need them.
The tents and bedding Lion King provide are adequate, but if you feel the cold and are going near Ngorongoro Crater think about taking your own blanket / sleeping bag, it can get cold. Have a look on their website, it's an accurate list of what goes with you in the car.
Hope that helps.
Mush 'n Kit