Dear Brynja
I have camped outside various lodges and on Uganda wildlife authority campsites in Queen Elizabeth, Murchison falls, Lake Mburo, Mount Elgon, Lake Bunyonyi, Mabira, Lake Nabugabo and Jinja. I have always felt very safe (although my friend did have all her shoes stolen when she camped in Mpanga!) it can be surprisingly cold at night, so do take an extra layer. Don't forget to take water too. camping facilities are very, very basic and may include a pit latrine toilet only. If you book in advance however, sometimes local food is available or a fish can be caught :)
if you don't have camping equipment, you can buy it when you get Kampala. GAME supermarket sells a range of tents, sleeping bags, etc at very reasonable prices.
Happy Camping :)