Ah, so when both are plugged in the wifi works but when the wire gets pulled the wifi stops working too.
So you can try this...
command: netctl list
see if the wlan0 is selected. if not, "netctl enable wlan" I believe is what creates the link for the service and then try a reboot.
If that doesnt work, what has always worked for me in this problem (and it happened with Volumio), I needed an ethernet loopback plug. They are easy to make just google for one.