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Is this doable
It is. But, it will not be easy or reliable (the WiFi reliability that we expect from our PCs is not to be found on the Arduino).
It would be far simpler to connect the WiFi router to the same router that the existing Arduino with Ethernet shield is connect to, and let the Arduinos exist on the same LAN, with some of them connected to the LAN wirelessly and one (or more) other(s) connected with wires.
The data moving back and forth won't know that it did, or did not, travel down a wire.