My take on this would be
- Central Arduino with ethernet shield and local mass storage like SD for card info and logging.
- Door Arduinos with some form of comms link to the central one.
Having an Arduino at each door is probably an overkill (but it could be one of the small clones) but required I think to localise the card reading.
If there aren't many cards the lookup table to match cards to doors could be stored in each door Arduino to reduce response time. In this case the door just tells the central processor what has happend.
Rob