I use paper bags, size determined by size of chameleon. They are not abrasive like cloth bags, and they aren't hard so the lizard can't get bounced off the walls, like deli cups. A wadded up paper towel provides something for the cham to hold onto.
Too much room = chameleon bounced around in transport.
take a paper punch for notebooks and punch some holes in the bag for ventilation before putting the lizard in there.
Roll the top of the bag sack-lunch style and clip it so the lizard cannot get out.
Tape it securely inside a cooler to the walls so the bag cannot move/slide/bounce/swing during transit.
Keep things dark- if it's a styrophome cooler put it in a cardboard box. Close the lid and open every few hours for air exchange.
Don't leave the cooler in the car unattended. It's surprising how hot how fast a car can heat up.
If you tape your chameleon bag up high, you can put a juice jug of cold water with a couple ice cubes in it on the bottom of the cooler where it won't touch the bag to help with temp control. Just make sure everything is secure so there is no possibility of the bag coming loose or the cooler overturning and crushing the cham with the jug. If your ac in the car works, there is really no need- just put the cham in a dark part of the care near the front. The cooler will keep the ac from being too much for the cham.
Darkness = chameleon falling asleep = least stress. Cooler = stable safe temperature.
I would not allow my chameleon to have access to light during the trip or too much room where it can get bounced around. Both = plenty of stress.