Here comes the insane idea.
The two brown lines are the bases of the mousetraps. They will both start in the loaded position, with whatever release mechanism you are supposed to use for the first. The second is positioned so that when the first is done, it will strike the edge of the second mousetrap and set it off.
The drive gears are set into slots in the frame, so that when tension is on a string, the drive gear attached to that string will engage the axle gear. The slots are the orange lines in the picture.
The wheel gear is the type with teeth around the outside, and another, smaller ring of teeth on its side. If you can't find one like that, glue a big and a small gear together on the axle. The first drive gear is set up for approximately 1/3 of the size of the larger axle gear. The ideal setup is for the gear to accelerate the axle at a slightly lower speed than what would cause the tires to spin. The second gear is approximately a 1:1 ratio to the smaller axle gear.
Here's how it should be set up, to clarify the picture. The first mousetrap should be set so that the string is initiall perpendicular to the mousetrap arm. The string should be short enough that the first drive gear is already meshed with the large axle gear. This will provide optimal torque to launch it off the line. The string should be tied around a pin on the axle, so that when the first mousetrap is almost finished pulling, it will be pulled clear of the axle, thereby preventing it from slowing the car or fouling up the geartrain. When tension on the string is released, the first gear should fall down the slot, clear of the axle. The release point should be when the string is at a 10 to 15 degree angle to the mousetrap arm. After the release, the arm should continue about another 3/4" to pull the string clear of the mechanism. At the end of this 3/4", it should strike and set off the second mouse trap.
The second gear should be weitghted so that it will not bounce and strike the axle gear before it is supposed to engage. It should have no more than 3/8" to move before meshing with the small axle gear, and some sort of blocking mechanism (such as a pin between the teeth on the side opposite the axle) should prevent it from spinning before it engages the axle gear. The gear transfer should probably occur at about the ten foot mark. Once again, the string should be attached to a pin on the axle so that it will disconnect and allow the gear to disengage when the string is 10 to 15 degrees from parallel to the mousetrap arm.