Hiya,
First thing I would check - is the SD Card fully inserted? It hangs out a bit, but it does need pushed in all the way to the connector. The boards should also be flatly connected with the tan 32-pin connector between the TinyZero and SD Shield.
When you connect the SD card and open it on just your computer, I assume that works just fine? Can you drop files onto it? Make sure to safely eject it when you unplug it - some data corruption could cause issues when trying to read the info from the card.
Let me know if that helps!
Cheers,
Réna