The problem I have with a PLC for this application is the limited amount of data storage available, especially in the low-end models, which is all you really need based on the amount of I/O and the lack of complexity.
How much research have you done into the capabilities of card reader systems? I don't know a thing about them myself.
I'm thinking that a simple viable system would be something that can both read from and write to the card, like an RFID system. The card data would need to include something that identifies it as one of your cards, an expiration date, the last date/time it was swiped, and a number between zero and five.
Each time the customer swipes it, the system checks for validity and the "swipes remaining value greater than zero, then checks the date/time of the last swipe. If the last time was today, deduct one from "swipes remaining" and write back the current date/time and the new "swipes remaining" value. If the last swipe was not today, write back the current date/time and "swipes remaining" equal to four. You might not even need a PLC.