I've heard of the following method being used before for fishes in a tank, so not too sure how it's going to work for your chinchillas, but anyway... what the guy did was put a piece of glass in the tank, restricting the movement of the fishes to a single plane, ensuring sharp focus. Perhaps you can try that....or maybe you can just try snapping them when they are eating or something...
First of all, know where you wanna shoot them. If you intend to do it in the room and feature them moving freely in the room instead of a caged environment, then set your camera there. Try to keep your camera in a low angle position. Chins are very fast creatures like what you said so relying on shutter speed to freeze the subject is near impossible unless you are out in the open. Use flash to freeze them. How many flash you can use depends on your budget. One flash can do the job, but in my setup I prefer 3 flashes.
You need a PVC pipe or anything that you can use to guide the chin to your camera. Position the pipe perpendicular or diagonally to your camera angle depending on what you want to feature on the Chin. One flash to light the background, I would use a blue gel over it and dim down the exposure so it wouldn't be too bright. One to light from the front of the Chins face. another to light the Chin just somewhere behind to create a rim light around the Chins furs. You can use a slight warming gel to light the Chins furs but it is not necessary. Ignore shutter speed setting as you will be using flash to light the whole thing. Set it at 1/60 sec or faster until X-Sync and aperture of f/8 or smaller to get sharpness on the eyes and nose. I would position the camera at a low angle and shoot upwards with a moderate wide angle lens to feature the room environment. Refer Megaweb's shot for a starting reference. Do a few test shots to make sure the exposure is right. If you are using 35mm film SLR then you can't test unless you don't mind wasting a roll to rush down to the lab for process before the actual shoot.
If you are firm with your setup, bring the Chin out of the cage and insert it into the PVC pipe. Wait for it to come out to where your camera is and snap. You need to prefocus your lens before doing that. As Chins are living creatures, you will have to try it a few times just to get it right.
Please note that I have never shot a Chin before as I don't have one but this is exactly what I would do if need be. This same technique was employed by myself to photograph snakes in Pulau Ubin and some other small creatures. You may scare the creature but they won't die. Even NatGeo uses the same technique when photographing gliding/flying snakes in Borneo. They push the snake through the pipe, blow it out to get the shot of the snake in flight before it lands on a net below. Then bringing it back and do the same thing again and again until the shot is confirmed.