Oh my, the dreaded disappearing sock trick has struck again.
There is a technique that some people have used. They partly open the door then press down pushing it from its snap in mounts.
Then reaching up and snag the card. Look at the door's relative position so you know about where the snaps are to place it before you reach under and fit/push it back up and into its original position.
Before you try this, please search the forum more or search Youtube and locate a clear description of how. But that's the basics as I recall it described.