A basic schematic would help. For example, I'm not sure how the reed relays fit in there.
1) a) The USB power option is provided primarily as a programming/setup convenience. IMHO, it is preferable to use the Vin or +5V pins for "permanent" power arrangements. (Which pin you use depends on what voltage is your power source.)
b) The Arduino board is not meant to SUPPLY POWER to anything, only to CONTROL it. Power the RFID card reader and the relays and the display directly from an appropriate power source and NOT through the Arduino board.
3) The FOX board seems pretty competent all by itself to talk to the RFID reader and the display and the lock solenoid. It is not clear why you even need an Arduino here?
This is a common RFI (radio frequency interference) problem - move the FOX board well away from the RFID reader, its putting out loads of 'hash' (wideband digital RFI). Your USB cable has ferrites both ends? If not its probably just piping the RFI direct to your arduino.
Not clear why you need more than one RFID per locker?
Thats a problem here is that we don't see the whole picture so we can't tell what your design objectives or tradeoff decisions are. How far away from each other are the lockers? Are they metal or something that can form a natural Faraday shield to protect them from interference from other RFID receivers?
What is the startup time for an RFID reader? It is practical to "multiplex" them?
Using multiple Arduinos and USB and hubs just seems unnecessarily complex and overkill. Extra development complexity, extra component cost and no apparent system benefit? Unless there is some other factor you didn't mention here
It is practical to "multiplex" them?
If you have to cover several modules, that could mean that the response time could be several seconds for any given module. I would carefully consider this factor before proceeding. It seems like it could be a show-stopper depending on what kind of performance you need
You still have not presented any design factors that would convince me that you need ANY Arduinos
why is the Fox board significantly reducing the read range of the RFID scanner,
Ripple on the supply to an RFID reader will severely reduce the range. As will running it under voltage. Are these 5V readers being connected to a 5V source or are they 12V to 5V readers in which case you are operating them at the low end of the range. An RFID reader will work best at 12V rather than 5V.
This tutorial is probably one of the most-cited as a reference for helping people with the kinds of symptoms you are reporting...
Does the voltage of the cap matter at all ?
and I read somewhere that you should try a cap equal to one over triple the crystal frequency,
fit decoupling capacitors to the RFID reader
I don't want to risk a trial-and-error approach to this and blow something up