If I or my opponent has a trigger in their ending phase, and I react to that trigger with Thrill of the Hunt, what happens? Will a showdown begin after the chain clears, or will the game simply continue to the next player's turn since we're already in the ending phase?

Partly — you can absolutely cast Thrill of the Hunt there ([Reaction] is legal in the Closed State the trigger creates), but no Showdown begins during the Ending Phase. The battlefield you play the unit onto becomes Contested immediately, yet a Showdown only opens during a Cleanup "in a Neutral Open State," and a structured step like the Ending Step isn't one — so it gets Staged instead (Rule 323.11; same logic as FAQ #9254 for the Beginning Phase).

The turn then finishes normally (Expiration Step: heal all units, "this turn" effects expire, rune pools empty) and passes to the next player. The staged Showdown doesn't evaporate, though: it stays staged while the battlefield is Contested with your units present, and it opens at the start of the next player's Main Phase — the new Turn Player picks which staged Showdown/Combat opens, and Focus goes to whoever applied Contested (you), not the turn player.

One distinction worth knowing: if you play that unit to a battlefield where your opponent already has units, it's a Combat that gets staged, not a plain Showdown, and it opens next turn the same way with you as Attacker. If you play it to a battlefield you already control, nothing becomes Contested and nothing happens at all.