Yes, you can. You need to find an Arduino shield (or other board) that lets you attach SD cards to your board, and use that. (A quick Google search on "arduino SD card shield" reveals this SD card reader shield, for example. There are also "breakout boards" that are not shields. You'd have to attach wires to those rather than plugging a ...
I figured out that one get these kind of problems (reader get´s firmware version but can not read tags) are related to power consumption. I tried to power the reader with Heltec Lora 32 Dev board and it schowed exactly this behaviour. Now, connected to another 3,3V source everything works as it should :)
I'm just starting a project using RFID tags to track cars and engines on a model railroad.
Although I haven't gotten very far yet, my first thought was to use a multiplexer to select which reader I receive data from at any given time. It won't be simultaneous but I should be able to read a dozen or more per second which is probably enough.