Step 1: Tools/Materials
-band saw/jig saw/coping saw
-miter saw/hand saw
-drill press - a hand drill alone will not work; holes need to be perfectly square
-wood drill bit [6mm, 7/32", 15/64"] - may vary
-wood drill bit [8mm, 5/16"]
-combination wood file
-drum sander/belt sander/palm sander/random orbit sander
-sand paper [100 grit] - for the powered sander
-sand paper [300 grit]
-sand paper [600 grit]
-drafting tools [pencil, rule, compass/2D CAD software, printer]
-metric hex keys
-router bit [45°]
-small wood block [2" x 2" stock, 1.5" x 1.5" actual] - I chose poplar as it was the smoothest I could get.
-4 inline skate axles [8mm diameter]
-2 bolts [M6 x 1.0mm x 50mm] - may vary - you can use threaded rod if building multiple cars
-4 standard inline skate wheels [any diameter] - must accept standard 608 bearings
-8 bearings 
-4 washers [8mm, regular or split]
For the wood, I went to Lowes and picked out the smoothest length of poplar I could find. They have a tiny section in the back of the store that has nice wood stock, in Pine, Poplar, and Oak. The wood in this section is overpriced, but one car doesn't require much. If you have an actual lumber supplier, go there and pick out something nice instead. Also, do you research and make sure the wood is food-safe, since these cars are intended for young children. Obviously, don't use treated lumber or any kind of particleboard.
The tires for the car are actually inline skate wheels, with the standard 608 bearings. I got mine from ebay, and they came with bearings. Most wheels, however, will come without bearings. You will need two bearings for each wheel, and can find them here. The bearings won't be supporting any substantial weight, so you don't need to use expensive, high precision bearings.
One of the problems that I found with Make's car was that the axles consist of hex bolts and are secured with lock nuts. This leaves big chunks of metal sticking out from the sides of the car. If kids are rolling these around the house, there is a big potential for scratched walls and table legs. Solution: use 8mm inline skate axles and a threaded rod as the axle. Many inline skate axles feature flat heads with Allen slots. This leaves a very small profile and no sharp edges. I got my axles here. Yes, I have extensively searched the usual online suppliers(McMaster-Carr, MSCDirect, Grainger, etc.) for sex bolts of the proper dimensions, but I couldn't find what I needed. Yes, I could machine my own bolt barrels and cut flathead slots in the heads, but the inline skate axles saved a lot of time.
For the threaded rod, I took an axle to Lowes and found that the axles are threaded at M6 x 1.0mm. This may be different depending on what axles you use. Because big box hardware stores in the U.S. don't stock metric threaded rod, I bought two M6 x 1.0mm x 50mm bolts and then removed the heads with a hack saw. On a side note, can you believe that Lowes no longer carries tap and die sets? They also suck at stocking their nut and bolt drawers; pretty soon I'll have to order everything from the internet.