Note that if it has a green LED in series with the optocoupler (that one does not IIRC), you need to connect the optocoupler input between an ESP pin and 5 V, not 3.3 V.
But you do not need so much to have an entirely separate supply for the relay coil, as to wiring those connections as a pair directly to the 5 V power supply output terminals, and keep the control connections running also as a pair back to the Arduino or ESP.