I have a small robot toy which can walk when I use the wire remote controller. How would I go about converting this to a wireless remote control for the robot toy?

That depends on what kind of control signal is being sent through the wire from the controller.

If the signal is digital, then it is being encoded by the remote, and decoded by the robot, then piped to control circuits. You would need to put a modulator and RF or IR transmitter on the remote, and a receiver and demodulator on the robot.

If the signal is analogue, multi-wire, or both, then it's more complicated. Especially if some of the drive/controller electronics are in the remote, and the robot is nothing but motors. You might be able to encoded/decode an analogue signal, but it gets harder as you basically start needing to replace internal circuits with your own, instead of just adding things.

Another option in this case would be to attach the controller onto the robot, and make your own remote which will control the original controller.