1994 Ford Explorer 4 wheel drive button will not put it in

Most likely, the motor on the transfer case is the problem. There is an electric motor on the rear of the transfer case that does the actual shifting. Through a set of contacs and switces in the motor, it closes a contact that lights the 4X4 light inside the car. There is also the possibility of a bad pushbutton. This is a less freguent problem, but is also a possibility.