Oh, right, you have to get them wet for them to be all soft and squishy. It’s been a while since I’ve used one.
Try the Duden. It is vastly overrated (it says in this entry that the origin is uncertain, for instance, which it is not: see chamois!), one of my pet peeves is exposing their shortcomings. So there is another one! Yay! And it is only included as an adjective, then as a compound part of the tanning process, not a noun on its own.
It does not even state the first recorded use! Another example of sloppy dictioneering. 
Yep: Sämischleder for the cloth, but Gams for the animal.
It is complicated, it seems. For this species the denomination is goat-antelope, and Wikipedia redirects goat-antelope to caprinae and goes on to write:
The subfamily Caprinae,[1] also sometimes referred to as the tribe Caprini,[2] is part of the ruminant family Bovidae,[3] and consists of mostly medium-sized bovids. A member of this subfamily is called a caprine.[4]
Prominent members include sheep and goats, with some other members referred to as goat antelopes. Some earlier taxonomies considered Caprinae a separate family called Capridae (with the members being caprids), but now it is usually considered either a subfamily within the Bovidae, or a tribe within the subfamily Antilopinae of the family Bovidae, with caprines being a type of bovid.
The nomenclature has changed, my interpretation may be antiquated. It may be evolving still, see the use of “mostly”, “some other members referred to”, “some earlier taxonomies”, “now it is usually considered”. Scientific clarity this is not.
Still, a very nice thread and an oportunity for Klugscheisserei (a blatant display of know-it-all-ness) I could not let pass 
I for one learned a lot of things in this thread. Firstly, that the first word in “St- Dominic’s Preview”, a song I dearly love, is “chamois”, that this is a cloth made of leather from the animal called “Gams” in German, that there seemingly isn’t an exact translation of “Gams” in English, but this animal is somehow a goat-antelope, and the German (rather Bavarian) word for chamois, “Sämischleder”, which I never heard before. But then, I don’t clean windows with leather cloths, never wash my car by hand and I’m not Bavarian.
I call that three out of three points for you. 
When i learned about that animal, it was called a type of goat.
You got me thinking about it and I’ve seen some smaller ones sold for cleaning eyeglasses. I had a ski jacket that had a chamois-like glasses/goggles wipe cloth sewn into a chest pocket for a similar purpose.
I notice that in addition to the silent q, one of the four m’s is invisible.