Can I get a wired (ethernet) connection off a wireless network?
The product you want is called a “Wireless Ethernet Bridge”. It’s more or less the same hardware as a wireless router, but it acts a bit differently in bridge mode. As folks above have noted you can take a regular cheap wireless router and flash some third party firmware on it to do the same thing. Be aware that it doesn’t work very well if the wireless isn’t reliable. I’ve had a lot of trouble with an old Slimp3 (Squeezebox) hooked up this way; it seems to not handle the idea that a wired ethernet connection could fail. I also had a lot of trouble until I switched from DHCP to static IP addresses behind the bridge.