We travelled around Iceland with public transportation in 1980 (!), at least in the summer there are 4x4 busses which drive also the inland routes like Landmannalaugar. I do not know about September, though, but the ring road should be doable at least.
I suppose camping is quite OK more or less everywhere like in all Nordic countries, but there are places where new tracks stay visible for decades (moss fields on lava sands), if not centuries, so be considerate.