How far apart are the readers going to be? You can use the 1024 - PhidgetRFID read/write at each of the locations, but it has to connect via USB to a computer of some kind (PC, laptop, SBC, raspberry pi, etc). USB cables can only reach about 5 meters, so you may need multiple computers if the locations are far apart.
The data recording can be done via software- you can use the TagRead event and dump the tag data and timestamp to an excel spreadsheet (there are many tutorials online for .xls and .csv output for whichever language you end up using).