"Club card" refers to the High Roller Club.

To get into the High Roller Club (and earn a Club card), you need to accumulate High Roller Crown points.

There are several ways to earn Crown points required to obtain a High Rollers Club card - By claiming the free coins at the store, spinning the bonus wheel, playing the vault bonus, leveling up, etc.

You can also find this information by tapping on the Crown icon on the lower left of your game screen.