Can I ready Irelia during Showdown when targeting?

Yes, you can ready Irelia, Fervent during a showdown when targeting her, provided you have an effect that allows you to do so.

Interaction Breakdown

When you play a spell or ability that targets (chooses) Irelia, Fervent, her triggered ability ("When you choose or ready me, give me +1 Might this turn") is placed on the chain. If that same spell or ability also readies her, or if you use a separate effect to ready her in response, she will trigger her ability for each event.

Key Rulings and Mechanics

• Triggering on "Choose": According to RiftJudge FAQ #8823, choosing Irelia as a target for a spell triggers her ability, granting her +1 Might. This stacks with any other triggers she receives in the same turn.
• Triggering on "Ready": According to RiftJudge FAQ #8590, readying Irelia triggers her ability, granting her +1 Might.
• Distinction between "Choose" and "Ready": It is important to note that not all effects that ready a unit count as "choosing" it. For example, the Blade Dancer (Irelia's Legend) ability readies a unit, but it does not "choose" or "target" that unit, so it does not trigger the "choose" portion of Irelia's ability (RiftJudge FAQ #8590, #6528).
• Stacking: If you use a spell that chooses her (triggering +1 Might) and then use an effect to ready her (triggering another +1 Might), she will gain a total of +2 Might for the turn.

Sequence during a Showdown

When you play a spell that targets Irelia during a Showdown, the following sequence occurs:

1. Play the Spell: You play a spell and choose Irelia as the target.
2. Trigger: Irelia’s triggered ability ("When you choose... me") is added to the chain on top of your spell.
3. Resolution: Because the chain resolves in LIFO (Last-In, First-Out) order (Rule 336.1), Irelia’s ability resolves first, granting her +1 Might.
4. Spell Resolution: Your spell then resolves, applying its effects to Irelia.

If you then use an effect to ready her (e.g., in response to the spell or after it resolves), that readying event will place another trigger on the chain, which will resolve and grant her an additional +1 Might.

Note: I could not fully verify the ruling regarding whether a single effect that simultaneously chooses and readies a unit triggers the ability twice.