Can I ambush during a showdown at reaction speed with Rengar Trophy Hunter if I am not involved in the showdown in a three player game?

No, you cannot play Rengar, Trophy Hunter to a battlefield that is currently in a showdown between two other players.

Reasoning

1. Unit Presence and Showdowns: According to RiftJudge FAQ #5108, a battlefield that is already hosting a showdown between two players is not a legal destination for a third player's unit. The rules prevent units from three players from occupying the same battlefield at once.
2. Legal Play Location: While Rengar, Trophy Hunter's unique card text allows him to be played to a battlefield with enemy units (even if you control none there), this ability does not override the fundamental game rule that prohibits a third player from joining an ongoing two-player showdown at that location.
3. Ambush Keyword vs. Unique Ability: As clarified in the retrieved FAQs (e.g., FAQ #9992), Rengar, Trophy Hunter's ability to be played to an enemy-occupied battlefield is a unique permission distinct from the Ambush keyword. However, this permission is still subject to the general limitations regarding board state and player counts in a showdown.

In a three-player (FFA3) game, there are three battlefields (Rule 482.4), but the restriction on interfering with an existing showdown between two other players remains in effect regardless of how many battlefields are in play.