There are cables coming off the headrest controls. Each movement corresponds to an input in the chair's control unit.
This isn't something that can just be hooked to a control since the "sticks" will be the headrest units.
The radio will have to be dismantled, the sticks removed and input jacks wired to each circuit going to the radio output. Where the right stick would go left to turn left, it would now be the left head rest being pressed left to give that signal to the Tx to send. If pressing the headrest farther makes it turn more, that's another issue that needs to be addressed.
Not something we can do here without the radio and chair. Once you explain it to someone locally who knows electronics, he'll be able to do what you need.
You might even consider the chair supplier since he would already have the parts in stock, except for the radio transmitter.