There are a number of problems. First, no connection to the EZIO2X4 Gnd should exist. This is not a Green wire ground connection. The Green wire ground to the EZIO2X4 is accomplished through the ground pin of the 3 wire plug connection.
Second, the DPDT relay coil is not being controlled by the EZIO2X4 Relay 1 contact. Wall plug 120v hot (black) to R1_B. R1_A to DPDT relay coil pin 8 (as indicated in your diagram). DPDT relay coil pin 7 to wall plug neutral (white). With this arrangement one side of the DPDT relay coil is always connected to the 120v neutral. When the EZIO2X4 relay 1 is energized it supplies 120v hot through its contacts to DPDT relay coil pin 8. This will energize the DPDT relay. When EZIO2X4 relay 1 is energized, the DPDT relay is energized. When the EZIO2X4 relay 1 is not energized, the DPDT relay is not energized.
Third, I think DPDT relay pins 5 and 6 connections to the wall plug are reversed. The diagram shows the motor hot/up and hot/down going through one set of the DPDT relay points ( 2 & 4) to 120v neutral (I was expecting to see DPDT relay pin 6 connected to wall plug hot (black)). There is no need to switch the motor neutral through DPDT relay points 1 and 3 but if you do, the motor neutral DPDT pin 5 should be connected to wall plug neutral (white). Motor neutral (white) can be connected to wall plug neutral (white). The 120v neutral circuit is normally not switched.
Make the changes to the EZIO2X4 Gnd, and the DPDT relay coil connections. Do not have anything connected to relay pins 5 and 6. Verify that when the EZIO2X4 relay is energized the DPDT relay is energized, when the EZIO2X4 relay is not energized, the DPDT relay is not energized. When that is confirmed working, then connect relay pins 5 and 6 to the appropriate wall plug hot and neutral connections.