Without knowing anything about the readers and tags, it will difficult to design anything that can read the tags. There are more than a dozen RFID standards that are in widespread use - all of them are different. Some of the various parameters are: operating frequency, passive or active tags, reader/interrogator modulation type and data format, tag response (active or passive backscatter), tag modulation and data format, transaction protocol, etc.
Operating frequency can be anything from 130kHz to 2.4GHz or more. Modulation types can be simple ASK, or more complex. Data encoding could be simple NRZ or some Miller, Manchester, etc. type encoding.
Reverse engineering your system is possible with the right equipment - but very time consuming. Unless the manufacturer can provide you with the system's details, it's really not practical to build your own reader.
If the tags are hard-coded (as opposed to being programmable), then it's possible that the unique ID of the tag might be etched or printed somewhere on the tag itself...