In my experience, USB-C external devices can be more difficult to power via an iPad or iPhone's USB-C port. Lightning external devices pretty much work as intended. It may be that that particular card reader is demanding more power than the iPad is capable of delivering to it. When this happens, the external device will simply not operate correctly & the device that it's plugged into will not see it. In this case, an externally powered USB-C hub will help, although it adds to the clutter and requires it to be plugged into a USB-C power adapter. An example:
Powered USB-C hub
If it is in fact a genuine Apple reader, it *should* work as intended without a hub. However, it could also be defective (this does happen).
Have you verified that the reader works correctly using an altogether different USB-C capable device (if you have one)?