Sorry, I'm very new to this...
Is there a possibility to connect this Data Logger Shield to an Arduino Nano. How would this be possible?
It specifically says that it is a shield for a Wemos D1, which is not a Nano, so no. It could be wired to work with a Nano, but it would then be, by definition, not a shield and, since there are shields available to plug into Nanos, it would be a particularly stupid idea. It is also rather expensive...
Try ebay 253122741975, and check the suitability for your Nano. I imagine there are others similar.