We moved from Denver, CO to Charlotte, NC. We had Mom, Dad, and 3 two day old babies traveling with us. We got a cage that almost looked like a rabbit cage, all wire. It had no levels or anything. We put them in the middle of the seat with a towel covering all sides but the one facing us to keep the sun off them. We made very sure to have it nice and cool in the car!! They did not even care as to what was going on and they all did fine.
Shipping: Hmmm, I would not do it! There is no guarantee of climate control. I shipped one of my cats once and they lost her for almost 2 days! Come to find out, she had actually been shipped to Europe and was left on the tarmack for more than 12 hours!!! She came home dehydrated but did ok. Chinchilla's are way to sensative to something like that happening to them. Best wishes!!!