As best I can tell, Amazon Air does not hold its own air carrier certificate; they contract with cargo airlines to operate their flights. Per Wikipedia, those contractors are Air Transport International, ABX Air, and Atlas Air. These operators then use their own callsigns.
For example, the aircraft shown in your first link, N1997A ("Amazon One"), is apparently owned by Titan Aviation, dry leased to Amazon Air, and then operated by a Titan sister company: Atlas Air. The aircraft bears the legend "Operated by Atlas Air Inc" on the paint job. So for the flight it happens to be flying to Charlotte at this moment, it would be identified as GIANT 3581, "Giant" being the codesign for all of Atlas' operations.
But another aircraft in their fleet, N331AZ, is contracted to Air Transport International, and uses the creative callsign "AIR TRANSPORT."
So they don't have an ICAO code or callsign; the companies they hire to operate their flights do.