This is up to you as the DM
The accepted answer already gives you the technical result: the poison damage is coming from an ability of the monster; nothing to be gained here. Move along.
However, you are not only asking for the technical read, you are also asking how to handle it in game:
In-game justification
Many races, including drow, have supernatural abilities, for example the drow magic (page 24 of the PHB) trait that allows them to cast spells like dancing lights. If you as a DM are concerned about the believability and verisimilitude of your world, you could say that drow elite warriors have a supernatural ability that applies poison to their attacks.
Considering your desire to potentially reward the players for defeating an elite drow without unbalancing the game, the rules also do provide some amount of support and guidance.
Looting equipment
On page 11 the monster manual under equipment states:
A stat block rarely refers to equipment, other than armor or weapons used by a monster. (...) You can equip monsters with additional gear and trinkets however you like, using the equipment chapter of the Player's Handbook for inspiration, and you decide how much of a monster's equipment is recoverable after the creature is slain and whether any of that equipment is still usable.
The Players Handbook does not list the high powered elite drow poison, but it does list basic poison. So it would be within guidance to decide a vial of poison might be recoverable, in addition to the armor and weapons, or that one dose of poison remains coating the blade.
For basic poison, the rules state
Applying the poison takes an action. (...) Once applied, the poison retains potency for 1 minute before drying.
These mechanics indicate that you cannot just transport a coated weapon and expect the poison to remain potent, the poison would dry and lose its potency. The players may learn this the hard way, if they just take the poisoned sword along.
Harvesting poison
If you determine the poison is applied supernaturally to the weapon by the drow attacking, and you want to make it possible to scrape off the poison: there is a section on page 258 in the DMG about Crafting and Harvesting Poison:
A character can instead attempt to harvest poison from a poisonous creature, such as a snake, wyvern, or carrion crawler. The creature must be incapacitated or dead, and the harvesting requires ld6 minutes followed by a DC 20 Intelligence (Nature) check. (Proficiency with the poisoner's kit applies to this check if the character doesn't have proficiency in Nature.) On a successful check, the character harvests enough poison for a single dose.
While these rules seem intended more for naturally poisonous monsters, the elite drow warrior technically also is a poisonous creature, because they are a creature and their attack is poisonous, if the poison does not come from equipment that could be looted.
You could apply these mechanics to scrape off a dose of the poison from the short sword, in a way that is balanced.
Game balance
It is easy to see why by default there is no irresistible poison short sword or vial full of poison to be looted: a typical PC attack without magical or feat enhancement deals about 10 damage. A weapon repeatedly dealing 4d6 (average 14) plus Dex bonus damage, or a poison repeatedly adding 3d6 (average 11) damage to these attacks would heavily influence PC damage output, and potentially unbalance the game.
However, if this effect is limited to a one time benefit per elite warrior, and if applying the poison to the blade consumes an action, then there is much less reason for concern about balance issues.
In D&D drow have a long tradition of screwing player characters when it comes to loot
This goes back all the way to the first appearance of drow in first edition in some of the earliest modules for the game. There drow sported enhanced equipment and poison, both of which lost their potency when exposed to sunlight. Queen of Spiders, page 127:
Drow weapons and armor are usually crafted from an adamantite alloy of metal. All such items (cloaks, boots, weapons, and armor) do not radiate magic, but are magical in effect. They lose their power if exposed to sunlight. (...) Drow sometimes poison their weapons, and a -4 penalty applies to all saving throws against such.
Male Patrol guard: Level 2 Fighter; hp 9; (...) chain +1, buckler +1, shortsword +1, dagger +1, hand crossbow with 10 poisoned bolts
The game balance reason for this screw-job is easy enough to understand: Imagine if every single drow guard they defeat provided the party with the equivalent of four uncommon and rare magic items.
Since then, design has changed and instead of strenghtening weak opponents with overpowered equipment which invariably will end up in the hands of the PCs to cause problems, monsters have more inborn power. But echoes of this tradition remain.
