Does Rek'Sai Swarm trigger on a non-combat Showdown?

No, Rek'Sai, Swarm Queen does not trigger on a non-combat showdown.

According to the established rules and RiftJudge FAQs (#8827 and #9415):

1. Non-Combat vs. Combat: When you move a unit to an unoccupied battlefield, you initiate an "open showdown." In this state, there is no combat, and no units possess the "Attacker" or "Defender" designation.
2. Trigger Condition: Rek'Sai's ability ("When I attack...") is an Attack Trigger. Per the game rules, these triggers only occur when a unit gains the "Attacker" designation during a combat showdown.
3. Result: Because moving to an empty battlefield does not initiate combat, Rek'Sai does not gain the "Attacker" designation, and her ability does not trigger.

If an opponent subsequently moves a unit into that same battlefield, the open showdown ends and a combat showdown begins. At that moment, Rek'Sai will gain the "Attacker" designation, and her ability will trigger then.