Can I play Borderlands 2 GOTY Edition without the original game?

Typically, Game Of The Year editions of games are the base game bundled with most (if not all) DLC released up to the release date of the GOTY. They are typically packaged to get players who have not yet purchased the game into the full experience without having to figure out what DLC they need.

The Steam store page and the Borderlands Wikia page indicate that:

The package will contain:

• The main Borderlands 2 game
• All DLC included in the Season Pass:
  • Captain Scarlett and Her Pirate's Booty
  • Mr. Torgue's Campaign of Carnage
  • Sir Hammerlock's Big Game Hunt
  • Tiny Tina's Assault on Dragon Keep
  • Ultimate Vault Hunter Upgrade Pack 1
• Gaige the Mechromancer
• Krieg the Psycho
• The Heads and Skins included in the Collector's Edition
• The Creature Slaughter Dome
• The Gearbox Gun Pack
• One Golden Key
• The Vault Hunter's Relic

NOTE: The Edition will not contain any of the content released after its announcement, including any Head Hunter DLCs and the Ultimate Vault Hunter Upgrade Pack Two: Digistruct Peak Challenge.

Game of the year edition implies it is the base game with possibility additional features that came out later in the year such as DLC. So yes.