You have Leonin Arbiter with the static effect that players cannot search their libraries unless they pay 2 to ignore it. But Karn, The Great Creator ability for -2 is "You may choose". Does sideboard count as library? Then if so, I assume then Leoning Arbiter affects this?
No, Leonin Arbiter has no effect on Karn's ability. The library is a specific zone in the game; it is where you put the deck to start the game, and it is where you draw cards from. The sideboard is a completely separate thing outside of the game; it is normally used for switching cards into and out of your deck between games in a match, and only specific effects like on Karn can access it during a game.