I'm running a home-brew campaign for my family and I have them going into the underdark. They are fighting Drow and Deep Dwarfs but the settlements I made for them lack stone-working places and places to make poison. That made me wonder, do they just extract it from their spiders? Do they have some kind of deal with the demons to supply poison?
It is, so far, a mystery, but we know some things
The first information we got on Drow Poison from 5th edition is from the introduction to the drow in the Monster Manual:
Poison Predilection. Distilled from spider venom and the flora of the Underdark, poison can be found in abundance among the drow... Drow mages concoct a viscid toxin that leaves enemies unconscious.
It seems that while spider venom and other sources of reagents are necessary, there is also a process that the Drow mages know of. Of this process, we know only what the Dungeon Master Guide briefly mentions in the section on Posions:
This poison is typically made only by the drow, and only in a place far removed from sunlight.
So we know that whatever process is used to make the poison, it can be performed without sunlight. However, that is about it. I expected more information on the poison to be present in Mordenkainen's Tome of Foes which expanded greatly on Drow society, but did not mention the poison creation process once.
In summary, the creation of Drow Poison requires spider venom, some sort of flora from the Underdark, and preparation via some unknown process which does not require sunlight.
Assuming you're willing to accept old-edition lore as your answer....
Drow Knockout Poison comes from a Fungus
Based on old lore from 2nd and 3rd Edition, Drow Knockout Poison is derived from a fungus, while many other poisons they may use are taken from Underdark creatures.
Their favorite is a powerful knockout toxin used to capture live prisoners. Drow poisonmakers extract this toxin from a slippery, black fungus that grows like great slicks of oil in certain Underdark caverns. Various natural predators of the underground, such as scorpions, purple worms, and especially spiders, are other common sources for poison.
Races of Faerûn, page 37 (3rd Edition)
We also know that...
The making of this drow poison takes at least 4 days and a well-stocked laboratory.
It is a black, heavy, gummy substance, rather like molasses, and reacts to both air and sunlight. It will remain potent for a year if kept in a sealed packet but loses its efficacy 60 days after exposure to air (either aboveground or in the Underdark). It loses its potency instantly when exposed to sunlight.
The Drow of the Underdark, page 89 (AD&D 2E)
Fun fact, this wasn't the only poison they used...they have a bunch, but here's my favorite for giving everyone nightmares: "Calling." It was a magical poison made from Aboleth mucus and spider eggs. After you consumed it, the spiders hatched, ate you from the inside, then came boiling out of your nose and mouth in a swarm, attacking everyone nearby. (See the 3E version of The Drow of the Underdark, page 94-95)
Y'know, just in case your nightmares weren't creative enough already