I guess you're out of luck. You need to flash a new OpenWRT image in order to get WiFi up and running. And in order to do that for the first time you need one of the Ethernet interfaces.
I can assure you that the dev board is available. When I've first ordered my Carambola I ordered one module and one package of module+dev board, since I already had a feeling I might need it, plus it's a lot quicker to get started with something when you have the dev-board that's already implementing the most important interfaces.
I don't know if the pre-flashed image on the carambola supports USB to Ethernet adapters, but if it does that'd be your way of getting a new image onto the device. But you'd be better adviced to just order a dev board, or better yet order the bundle with dev board+carambola, it's not that much more expensive and you never know when you're gonna need that second carambola
That being said, the module tends to reboot upon WiFi activation if you don't supply enough power to it. I think the voltage regulators on the dev board were rated at 3A. But the actual consumption on the 3.3V rail is somewhere around 500mA. The exact numbers can be found in this forum (there's a thread on Carambola power consumption).
And if you think you've got a power supply that can handle this current already connected, check again. Some cheap switching power supplies won't really deliver the rated current, or at least won't deliver it for very long (the internal regulators might overheat, causing them to shut down for a few milliseconds, forcing the Carambola to reboot).