The web UI is getting all the mining stats from the cgminer API, not from anywhere on the filesystem. It connects to the cgminer API via a TCP socket (I think), requests the info, then displays it on the webpage.
You can use something like ncat to grab the info and write a script to grab the info and then use wget or curl to send it to the web service.
More info on the cgminer API:
https://github.com/ckolivas/cgminer/blob/master/API-READMEEdit: I'm not sure if the OpenWRT firmware on the AntMiner has netcat and/or ncat, but you can give it a try