Well as with many things, getting the live audio feed from the repeater working ended up being a fiddly process, but it was a learning experience every step of the way. Feel free to check it out on the right side of this page!
The current setup is a Raspberry Pi running Raspbian, with an older scanner feeding the audio into the Pi, and then being uploaded using DarkIce to the Broadcastify servers. This offers a pretty small physical footprint, with low power usage, and makes it relatively bulletproof. In the future I look forward to getting everything reconfigured utilizing a USB SDR receiver, which will eliminate the need for the external scanner and the required power, since the USB SDR stick will be powered off of the Pi itself!
Once I finally get all the little bugs worked out from this project, I hope to build up a second Pi with an SDR receiver that can be web controlled. But, we shall see!