802.11 wireless (WiFi) works in different frequency ranges than 802.16 (WiMax). Generally speaking the radios in the hardware will be limited to the frequencies in use by their technology, so you can't mix the two.
Since WiMax can operate in a number of different frequency ranges, the hardware designed to operate in one frequency range may not be able to work with equipment designed for another frequency. Some will be able to operate in all of them, so you will need to research your hardware.
One of the major differences, is that WiFi operates entirely within unlicensed areas of the RF spectrum. Many WiMax solutions (Clear/Sprint for example) operate in licensed RF spectrum. If the equipment is designed to operate in one of these licensed frequencies, you can only operate the equipment if you have a license to do so or are authorized to do so by a license holder as part of their service (i.e. you are paying them for service).
So even with the closed network, make sure you are not operating in any licensed spectrum to avoid potential legal issues and fines.