If I'm not mistaken, the top runners of the Any% Yes Man category (of whom I watch tomatoanus) run the game on the first released version. Is it still possible to obtain the first released version, or to unpatch New Vegas to the first version? I've searched and came up empty handed.

you're mistaken. you can run the game on the latest patch, and i'm sure almost everyone runs the steam version.

Idk if the original version was released on steam, but if it was then you might be able to download it with a steam depot. If all else fails you can try to look for an old torrent of the game, though I wouldnt do speedruns if you have a cracked .exe file if you cant find a legit one.