Taking apart an old radio control car is a great place to start. It has motors, a chassis and a battery already. All you need to do is find a processor of some kind and make sure your control signals work. While operating a radio-controlled car will not satisfy the requirements of the merit badge, taking an RC car, adding a controller (the computer brain that does the robot’s thinking) and some other mechanisms probably will. As always, talk to your merit badge counselor first to make sure that what you are planning is safe and will satisfy the requirements.
Many reasonably priced robot kits meet the needs for requirement 4. The robot kits available through the BSA (at Scout shops and www.scoutstuff.org) will clearly meet the needs of the Robotics merit badge requirements. A remote control car, plane, or boat will not qualify unless extensive modifications are made.