Take a small jug and cut it to shape and it appears they have mesh attached to the inside. You want to leave part of the bottom of the container and just cut a section out of the front.
Maybe a bat? If humans are to mammals, as storks are to reptiles (stork=bird=reptile). Then you would have to do the converse and have a mammal deliver the little chameleon babies, hence, a bat (the only mammal capable of flying). I keep imagining a tiny little bat carrying around a minuscule cham wrapped in a small white cloth. Kinda satisfying.