There was an exemption, but it was not for wine. From the Volstead Act:
The Volstead Act defines “intoxicating” as “any such beverages which contain one-half of 1 per centum or more of alcohol by volume.”
Why would this be unconstitutional? The US passed a law prohibiting the use of peyote as it does numerous other drugs. As it happened a religion already existed which had an established use of peyote in its religious ceremonies. Established for perhaps thousands of years*. As with the sacramental exception of wine during prohibition so too the Native American Church got an exception. If all anyone needed to do was form their own church and then say, for example, LSD was integral to their services you’d never be able to enforce any drug laws.
Maybe it has to do with a kind of repayment for the raping, pillaging, and killing of em, the years of abuse they took, the land that used to be all theirs, and some respect for the religion that was here before white man even found the continent.
Just an idea. 
RCers call it mustum (MOO - stoom), and it must (pardon the pun) contain a trace of alcohol for it to be considered valid matter for the sacrament.
Pax.
Jehovah’s witnesses do use wine. I attended a number of observances of the “Lord’s Evening Meal,” as it is referred to in Kingdom Halls. Everyone is offered the unleavened bread and the wine (real Jewish wine, I might add) but only people who are of the “elect” or “anointed remnant” actually consume it–the doctrine is rather complicated–and I have never seen anyone partake.
Considering that Indian nations are treated as foreign nations in the constitution, not at all.
Refrigerators as we know them now? Of course not… but check out:
Studies in Ancient Technology: Heat and heating; Refrigeration, the art of cooling and producing cold; Light by R J Forbes (Brill Academic 2nd rev ed. 1997). One of a series of books each containing a number of essays on ancient technology, covering both the pre-classical and classical periods. This volume covers heating, refrigeration and light.
Evaporation and digging of cold storage pits were used to some good effect.
This exemption applies off reservations too.
There is no exception in the Constitution for cases of white man’s guilt. Which unquestionably this exemption was. Congress threw the red man a bone. I’d say it is just a matter for someone outside the NAC to challenges this. Probably it hasn’t happened yet because the courts would throw out that part of the CSA, which would mean that not even the NAC could possess or use peyote.
Actually, then why can’t my church use marijuana? (Yes, believe it or not I’m at least halfway serious.)
That is my point. Why can the Native American Church use peyote to worship their god, yet I can’t use marijuana to worship my god? Is not a legal exemption for any specific church blatantly unconstitutional?
Your church has never used marijuana to worship its god whereas Native American religions used peyote for centuries, if not millennia before this continent was discovered, let alone this nation founded.
If you can’t see the difference between the two, I can’t help but think you’re being disingenuous at best.
As in “half a loaf…”? 
