Who gets to use a reanimated Gamekeeper's ability?
Your opponent. Leyline of the Void has no bearing in this situation, because the Gatekeeper is going to your (its owner's graveyard).


  400.3. If an object would go to any library, graveyard, or hand other than its owner’s, it goes to its owner’s corresponding zone.


When an object refers to you, it means its controller.


  Card Text: When Gamekeeper dies, you may exile it. If you do, reveal cards from the top of your library until you reveal a creature card. Put that card onto the battlefield and put all other cards revealed this way into your graveyard.
  
  109.5. The words “you” and “your” on an object refer to the object’s controller, its would-be controller (if a player is attempting to play, cast, or activate it), or its owner (if it has no controller). For a static ability, this is the current controller of the object it’s on. For an activated ability, this is the player who activated the ability. For a triggered ability, this is the controller of the object when the ability triggered, unless it’s a delayed triggered ability. To determine the controller of a delayed triggered ability, see rules 603.7d–f.