Wild camping in the region is generally not a problem, although there is always some kind of a security risk involved. All countries are sparsely populated and you can almost always find a place to camp wild without being spotted. Wild camping is not illegal (except, in a way, in Uzbekistan). Organised camp grounds in popular holiday spots exist. In Kazakhstan there are a few new, ridiculously overpriced places, in the rest of the region, you can stay for little money at a ‘turbaza’, a Soviet-style base camp with some facilities.
If you find yourself near human habitation, it’s never a bad idea to ask if it’s ok to camp nearby, this land might be used for grazing animals. Often times you will find yourself invited in, without the need to set up your tent for the night.
Kazakhstan is largely empty, so camping shouldn’t present too much of a problem. There is a population of around 30 000 wolves in Kazakhstan that you should be aware of. Camping gear like gas bottles, tents, etc. is sold in every major city. Dedicated outdoor shops don’t exist outside of Almaty and Astana, but bazaars tend to have a few stalls catering to the fishing and hunting crowd.
Camping in Uzbekistan is definitely tricky, because foreigners are supposed to register every single day in a hotel. Many people do camp, and get away with it. A solution to the problem is to have a night in a hotel every 2-3 days when you are in the cities, and should you be questioned about your lack of registration slips, say that you were in transit. Not a lie, if you are cycling or driving.
If you are not cycling or driving, I suggest to spare yourself the hassle and paranoia and simply pay for hotels. There is budget accommodation available for shoestringers in every big city.
In Kyrgyzstan, camping is still a matter of national pride, as the many yurts you will see in summer testify. You will have no problem finding a spot. Bishkek has outdoor shops, and camping gear is sold on Dordoi bazaar and the bazaar in Osh.
Just for completeness, in Tajikistan, wild camping is also not a big deal. Dushanbe is obviously your best bet of finding any camping gear.
If you get stuck somewhere along the road on your transit visa, camping is your only option on one of the long stretches of desert in between cities. If you are on an organized tour, camping near the Derwaza gas crater or the Yangikala canyon is a must.
In the relatively safe Wakhan corridor, wild camping is often your only option. In the rest of the country, there is a chance that you get murdered. If the police spots you, you will probably be moved to a chaikhana for your safety.
Obviously, Mongolia is even more empty than the rest of Central Asia, so space for camping is not an issue. There are several things you have to take into account that are specific to Mongolia, like extreme wind speeds, flash floods, forest fires, waste disposal, soil erosion and the value of pasture land. There is a great discussion over at the Lonely Planet forum with some expert advice to be heeded. You will be able to find most of what you are looking for in Ulan Bator in terms of camping gear.
About 10 minutes away from the sensibly dressed crowds of Chinese, you will find yourself alone and abandoned near Xinjiang’s tourist attractions. Outside of the well-known places, finding a camp spot should be even easier and much cheaper than the inflated prices inside the national parks.