Thanks - you talked me out of building that driver, which would have been a bad idea. I have 0 experience building H bridges - so starting with a smaller one would definitely make more sense. If I have to spend a lot on the motor driver, I may not even want to do it. I've read the Powerwheels motors are pretty inefficient.
They are fairly inefficient; first off, they are brushed motors, and secondly they are -big- (550 size, IIRC). But until you know the current needs for the motors, you won't know what to buy. I can tell you, though, that anything under 10 amps is likely going to be too small. You're going to want a driver (single or dual - pick yer poison) which can source at least 15 amps per motor; 20 amps per would be even better. Try to get a current measurement on one of the motors first, though.
To do this, remove the motor from the gearbox. Mount the motor to something immovable, and with a piece of leather or something around the output pinion gear (to protect it), clamp a pair of vice-grips to the gear, then attach the vice-grips to the bench or whatever you have so it won't move, either (you DO NOT want a spinning pair of vice-grips!). Find a 12 volt 20 amp power supply (an automobile battery charger with a starter setting is a good option - most of these can source 50 amps or so). Also get an el-cheapo multimeter (for both the battery charger and the multimeter - harbor freight is your friend, if you have one nearby - that or Northern Tool, or any other cheapo tool discounter) - don't do this test with a good meter.
Put the meter into the highest current measuring setting possible (like 20 amps), and hook the leads to the meter appropriately (there is generally a lead jack just for "high current" measurement - consult the manual). Hook the ground cable of the power supply to one side of the motor, then hook the negative lead of the meter to the other side of the motor. Turn on the power supply, then very, very briefly touch the other lead to the positive cable of the power supply. Hopefully it didn't arc-weld itself (if so, hopefully the meter burns out first!); you should see the current on the meter rise to a value - note it. Do this a few times, then compute the average value and add 10-15 percent; call this your "stall current", and size the h-bridge driver to that value (worst case scenario).
Note that the above method can be destructive to the motor, power supply, meter, wires, etc. Have a fire extinguisher handy, and wear goggles and maybe gloves. You might even have the power supply connected to a switched outlet or an extension cable, and a helper who can switch it off immediately if needed. If all goes well, you should get a good current reading and no damage. If the current needs are higher than the meter, the meter will blow (and if the motor is still good, you'll have to use the indirect method using a shunt).