In a Showdown, I cast a Reaction spell targeting an enemy unit. Since Ezreal, Prodigal Explorer has the REACTION tag, can I chain Ezreal's ability if the targeting condition is met? Can I also chain Vi, Destructive's ability on top of them?

Yes for Ezreal, no for Vi.

Prodigal Explorer (Ezreal's Legend): Yes — its ability has the [Reaction] tag, so it can be activated during the Closed State while your spell is on the chain. The "chosen twice this turn" condition still gates it: your Reaction spell targeting an enemy unit counts as one of those choices (targeting = choosing), so you need one other qualifying spell/unit-ability choose earlier this turn. Pay the exhaust cost and it resolves on top of your spell (LIFO).

Vi, Destructive: No — her ability ("Recycle 1 from your trash: Give me +1 might this turn") has no [Reaction] or [Action] tag. Plain activated abilities can only be used during your Main Phase in an Open State, not during Showdowns (Rules 145.2, 381), so she can't be chained onto the open chain — or activated anywhere in that Showdown.