can rengar go straight to a a battlefield from my hand without a showdown trigger

Yes — Rengar, Trophy Hunter doesn't need a showdown or a chain item to go from your hand straight to a battlefield, provided the battlefield has enemy units on it (or, for his normal Ambush half, units you control there).

Ambush is just a passive permission that adds a legal location and grants [Reaction] while he's being played to that location (Rule 822.1.b–c), and Rengar's own text expands it to "a battlefield where there are enemy units, even if you don't have units there" — so in your Main Phase, in a plain Open State with nothing on the chain, you can just play him there. That play applies Contested and a Combat Showdown gets staged in the next Cleanup, so the showdown is a result of him arriving, not a prerequisite.

The two limits: he can't be played to a battlefield with no units at all (empty battlefields aren't a legal Ambush location), and the enemy units must still be there at finalization (Rule 822.3) — if the battlefield empties before he finishes being played, the play becomes illegal.