according to this link, no camp sites are found in that area. I haven't heard of any as well.
Hotel Nais, just before Nish, used to have camp site, but I'm not 100% sure if they still maintain their camping ground, after this hotel was sold to private investors a couple of years ago.
see this link
Some rough distance between Hotel Nais and Leskovac would be 50km by the motorway, and some 45km between hotel and Prokuplje by regional Nish-Prishtina road.
If you're going to Nish/Leskovac/Prokuplje area from Budapest, the easiest option is taking motorway from Budapest via Szeged/Novi Sad/Belgrade/Niš. Except the stage between Hungary/Serbia border and Novi Sad, the rest of your route would be full profiled motorway to Nish and almost until Leskovac, and add 25km of regional two-lane road between motorway near Nish and town of Prokuplje.
The easiest way to continue to Zagreb and further on, is taking back the same motorway until Belgrade, and then continuing via Belgrade-Zagreb motorway. In this case, except some 30km just behind Serbia/Croatia border, you'll drive via full profiled motorways.
for distances' calculations.
You have to acknowledge that Leskovac and Prokuplje areas aren't the most developed parts of Serbia, so possible absence of good accommodation options shouldn't surprise you. On the other hand, there are many options in Niš, but still I can't recall any functional camp site.
edit: link modified