When we took our cham to the vets we took a cardboard box and poked a branch through the sides so it runs through the middle, lined the bottom with a towel, and my girlfriend held the box all the way in the car. In the beginning you could hear Nosy scratching around, but he soon settled down.
Here's a good link for info
http://muchadoaboutchameleons.blogspot.co.uk/2014/01/traveling-with-chameleon.html
The hard bit was getting him in the box

He was having none of it!
Hope this helps