Looking for the WoW Classic Armory? You're out of luck, I'm afraid - there is no Armory available for WoW Classic, just as it wasn't around in the original game 15 years ago. The Armory is only currently available for characters in modern World of Warcraft.
If you're unsure what the Armory is, it was first implemented into World of Warcraft back in the Burning Crusade expansion. It collects in-game information about individual characters - and guilds - and displays them on their own profile page on the official website. You can look up any character in World of Warcraft that is level 10 or higher and view their achievements, equipped gear, item level, talents, and raid progression. You can even check out which mounts or pets they've collected.
So why is the Armory not a thing in WoW Classic? It seems that, while the technology is present to implement it, Blizzard has decided that it would rather preserve the feel of the original game and leave it out. From the WoW Classic FAQ:
"If you wanted to know more about a player’s gear before you invited them to your party, you had to meet them in the world and physically inspect them (and you couldn’t inspect players of the opposite faction until 2.4). You couldn’t see their spec or talent builds when you inspected them, either. To maintain that same dynamic, we won’t have an Armory or profile pages for WoW Classic at launch."
While there's no Armory available right now, there is a chance it could be implemented in the future, depending on how Blizzard decides to handle the progression of Classic following the currently known content phases.
So, for the time being at least, if you want to check out a player's gear, you'll need to inspect them in-game or if you want to know their progression, you'll have to find out the old-fashioned way and ask them.