The two main items you will need for this are a momentary push button switch (this is the kind of switch that is only on/activated while pushed) and a relay. You will also need some additional wiring to connect everything, as well as the male and female connectors shown in the diagram below.
The momentary switches can be found at Radio Shack, or other stores that sell electronics. Here are a few links to some from radio shack
http://www.radioshack.com/product/index.jsp?productId=2062545
http://www.radioshack.com/product/index.jsp?productId=2062508
http://www.radioshack.com/product/index.jsp?productId=2062510#
The relays can be found at your local auto parts stores, but they usually run around $10-$12. You can get the much cheaper from All Electronics here
http://www.allelectronics.com/make-a-store/item/RLY-455/12-VDC-AUTOMOTIVE-RELAY/1.html
This wiring diagram created by 12vwiz shows what gauge wiring you need and how to connect everything. Wire gauges shown are the minimun size required. (click to enlarge)
The best place to mount the push button switch is on one of the spokes of the steering wheel in easy reach of your child's thumb. I would recommend putting on the side of your childs dominant hand (righty or lefty)
Here are some pictures where the wiring is internal to the steering wheel. The button used here is from a Power Wheel Lil Quad, and can also be found on some Power Wheels 12V Harley models and the 12V Kawasaki KFX Ninjas, and some other model that utilized a 12V "turbo" button.
Certain steering wheels don't have and internal structure like the one above, but you can still install the push button switch as shown here courtesy of blyndsyder
And you can also add wire loom to cover the exposed wires as shown in this image courtesy of ncdadof3
Here is another version shown added to the common Jeep steering wheel courtesy of drew510
Good luck and happy modding!
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