Actually there is what appears to be a USB micro-b port above the Ethernet port. If it's a beagleboard plugged into something else (which is what it looks like) there will also be a full sized USB-A port on the back for plugging stuff into.
I'd like to run this over wifi. I run my Avalon over wifi and I did the same with my old GPU miners. I'd been planning on using an old router and using it in bridged mode but if I could just plug in a USB wifi dongle from my old GPU miners, that would be nice. (I'm planning on cannibalizing a PSU from one of them anyway).
Also, you said an FPGA would be useless, but not really - you'd use it to send signals to the pins on the ASICs. The Avalon uses one. Otherwise you'd need to embed Ethernet, USB, PCIe or something like that directly into the ASICs.
Anyway, the board doesn't seem to be JUST a beaglebone, but rather a beaglebone plugged into some other PCB as a shield. The other PCB is probably some custom thing ORSoC cooked up. It could just be distributing the signal from the beagleboard or it could have an FPGA for controlling the chips. We'll see. Some simple FPGAs can be really cheap, like a couple dollars.