I transport mine in cardboard boxes. I put a branch or a vine in the box, place the chameleon in it and tape the lid closed so it is nice and dark. They usually respond by snoozing on their trip to the vet.
I took mine to the vet the other day and did it the same way as Heika. Box with a branch and darkness. He slept all the way there and all the way back.
if you live in a cold area you may want to put one box inside another and fill the space in between with bubble wrap, styro peanuts or other insulation. I transport Cyrus in a Styrofoam container lined with t-shirts and a heating pad underneath (I plug it in for about 1/2 hour before we leave then plug him in at the vet if we are going to be any length of time). Obviously if you live in FL or some other warm region this is a moot point.
I do recommend a t-shirt or other material so he is not sliding around. Don't use terry towels as their claws can get caught. I use my old, soft, worn t-shirts - works great
When I took Fractal the other day, the vet was really pleased and made comment to the way I brought him. I had him in a PVC rectangle covered with black mesh screen. All his vines/rees in place with all possible feeders and water bottle made to drip and catcher below. All done in light weight plastics and cardboard box bottom. He was completely at home in it. I covered it with two towels and then a sarong over those to keep out chill. Over that, half way down, a plastic garbage bag because it was raining. Everyone oohed and ahhed. lol