You could use tappable overlays, each card is in a loop so they are only on-screen for a certain amount of time and off-screen for the rest (so they don’t overlap).
Alternately, of you’ve already created the card loop, you could use “invisible” tappable overlays that cover the entire screen. One for each card and loop them on and off screen in time with the cards looping.
Once one of them is tapped, remove all looping overlays (since an infinitely looping overlay never ends), then just display the card they tapped on or whatever happens next in your story.