Reddit mentions of uxcell a11101800ux0165 DC 3V 1100RPM 0.2A High Speed Mini Vibration Motor for DIY Toys

What are the dimensions of the space inside the steering wheel?

https://www.amazon.com/uxcell-3100RPM-Vibrating-Vibration-Massage/dp/B00AKWRWRW/ref=sr_1_7?ie=UTF8&qid=1478586664&sr=8-7&keywords=vibration+motor

https://www.amazon.com/1-5-6V-50000RPM-Coreless-Vibration-Motor/dp/B00QC2F4FQ/ref=sr_1_5?ie=UTF8&qid=1478586664&sr=8-5&keywords=vibration+motor

https://www.amazon.com/1100RPM-0-2A-Speed-Vibration-Motor/dp/B0087Y95X4/ref=sr_1_3?ie=UTF8&qid=1478586664&sr=8-3&keywords=vibration+motor

but also if you can't find something small enough to fit into the compartment and powerful enough to shake the wheel you should consider putting several of the phone vibrators in parallel around the steering wheel itself.

Interestingly enough, I just pulled a failed Li-Ion battery from a small R/C plane that would be just about the right size for a rodent car. Image

So that's a 3.7 volt battery with a capacity of 250 milli-amp hours. We can fudge this number and say between both a slightly larger pack and advanced miniaturization, they use a 400mAh battery in a theoretical rodent-sized vehicle.

Now our drivetrain- A cellphone vibrator motor is about the size of motor you'd be able to use. It'd have to be a single motor with a gearbox driving the two front wheels, most likely, as I can't imagine managing to build a multi-motor drivetrain that's not going to fry your battery and they really don't need all wheel drive in Little Rodentia anyways.

So as we can see, that motor there uses 3v at 0.2 amps (200mA). With a good gearbox reduction, probably should be enough power to accelerate a vehicle that weighs less than 300 grams at a decent clip. But that's getting into a ton of force calculations that I reallllllllly don't want to do at the moment.

ANNNYWAY- we never want to draw 100% of our battery's capacity, because not only is the last 15% not usable due to low voltage, it's bad for the battery's longevity and a good way to overheat it and cause a short, leading to fires and death. Not good. So we should use, say, a 25% capacity buffer (Li-Ion should handle a 75% depth-of-discharge easily enough), so our 400mAh battery now becomes a 300mAh battery. Alright. That's how much energy we have available to us. Tack on a few milliamps for accessories and lighting, plus 5% conversion and discharge inefficiencies, and say there's about 280mAh available to the motor.

At "full" load, the motor should use 200 milliamps. In practice, it's going to use much more when accelerating (probably close to 275mA) and much less while cruising (maybe 100mA or so.) But as a running average, let's just call it a day and say the damn motor uses 200mA during normal driving. Yes, I'm a lazy engineer, so sue me.

That means the motor should be able to run for about an an hour and 24 minutes continuously. Again, fudge a safety factor in for heavy acceleration in city driving and other inefficiencies in the gearbox and wind resistance and other shit that I haven't totally accounted for, and let's just say you can get an hour of safe runtime out of your little electric rodent car. Sounds pretty good, yea?

Now, let's convert that to actual range. Let's pull some more numbers out of my ass, and say the tires on this car are 1cm in diameter. That's a circumference of 3.14cm, so the car can travel 3.14cm per rotation of the tire. The motor I linked has a free-speed of 1100rpm, but with a load on it (assuming the final gearbox stage is 1:1) it'll probably be closer to 1,000rpm. 3.14 time 1000 is 3,140cm per minute, or 28 meters per minute. In conventional terms, that's 1.88kph or 1.1mph. Being able to run the motor for about an hour means our equivelant range is also 1.88km or 1.1mi. That's probably not too bad in Little Rodentia, but for getting anywhere else that's a bloody slow ride to get not very far. And if you look at the scenes where we see rodent cars, both in LR and driving along the street where Nick is walking, 1.8kph probably isn't too far off from an average speed.

With this information, it's probably safe to say that there's no "one-car fits all" in Little Rodentia. our little theoretical car here would work just fine in LR which probably isn't even half a kilometer in it's longest dimension, but you'd need a different, much faster car for driving anywhere outside of LR. This makes the subway and/or pneumatic tube idea much more palatable for long distance transportation, since they can just zip along a nice, sealed tube at 40mph and be on the other side of the city in less than half an hour.
***
...Good lord that comment turned into a monster. Then again, I just got a new keyboard and love typing on it, so maybe that contributed to it's length... lol. I'm gonna go clean the house now.

At its simplest all you need is a motor with something lopsided on the shaft, which you could theoretically build from scratch. However there are some good quality low cost ones on amazon for a few USD under the uxcell brand, which I have had good experiences with (but with other products). Like this for example.

Also, don't mix too much hardener...if it's a 2 part resin, one part is the hardener...if you use too much, what happens is the mixture gets too hot and contracts which causes pitting/bubbles. This was the case when using Fiberglass Resin when I was making custom dashboards and trunk layouts for show cars. Use a little less than recommended, you'd be surprised. Keep it just a hair cool so it doesn't harden as quickly...also the vibrator motor idea as /u/akelis mentioned is smart as hell. Use a pager motor if you can, like this one here.