I am a total amateur at the cham stuff, but my first thought was to make boxes big enough to fit each of them in. If you have multiple chams, that is.
If you have 20 chameleons, make 20 little boxes. Twist a paper bag so it's a branch and then tape it to the sides/find a way to attach it. Also put towels under that. I'm guessing the box would be around 9"x9"xhowever tall they are.
Leave a feed cup in there. If disaster strikes, you'll have 20 little 'moving boxes' to put them in, and then perhaps even find a bigger box that ALL of them can fit into snugly.
That probably sounds like a terrible idea, but I really like the idea of an emergency moving box type of thing.
But I don't know what you're going through, and I don't know if you'll have any type of warning for this 'disaster'.
EDIT Reading other comments, it's a tornado, right?
Just grab the whole cage

I hope you only have one or two chams so you don't have to make tons of trips.