It's hard to find more than anecdotal evidence -- there's no set policy as far as I know or can find. This flyertalk thread however mentions
Some airports with a very sophisticated baggage handling systems have a minimum weight of what can be checked-in (don't know about size), I think it's something like 3 or 4 kilos minimum.
To the contrary,
I once checked a cardboard box that was about 8 inches long, 5 inches wide and 3 inches deep.
This quora answer
The young lady very sweetly obtained a carrier box and placed both a baggage label on the valise, as well as affixing one on the box she politely sealed up.
From my own experience: I shudder at the thought of an envelope going through the system. There are all sorts of belts and an envelope might just drop between two and get stuck... forever. This image actually shows a situation where it's patched up so it can't happen at this point but I wouldn't bet something obviously important to me that it won't. Also, it might simply drop off outside and get overlooked especially in bad weather.
I would rather ask for a cardboard box at any grocery etc similar shop drop my envelope it in, at check in I would ask them to put a sticker on the envelope and the big box so that if the big box gets destroyed your envelope still has a chance. Discard the box at other end, repeat coming back. Bigger is better. And I would still ask whether the light weight is a problem.
The smallest bag I ever checked in was actually a month ago when I checked in a Red Oxx Extra Small Aviator Bag as I didn't want to haggle with the TSA potentially about liquids and frankly, I didn't need more than 10kg of dirty clothing on 12+6 hours of flying. Also, it's small as duffels go but at 15"L x 8"W x 11"H it's not like an envelope, that's for sure. While this is not an ad for the XS, I am an extremely satisfied user of that bag, one of the best bags I own. I would even cautiously recommend buying that bag, drop a few things in it you don't need while flying and check it in. The cardbox box is cheaper that's for sure.