The AV rack has a connection to support formerly a Terradek and now a RazrRipsaw. This connection is both an HDMI output as well as a 3.5mm stereo (M) output connection to allow you to attach one of these devices. To answer your question directly, for originating a webcast. As an originating source of the webcast the audio does not go to the chapel but instead the chapel audio is sent to the encoder.
The customer panel has multiple audio outputs. 1. The Chapel Mix output, 2. The Organ output, 3. A set of outputs that are simply cable pass throughs from the stand for wired choir mics. I think the intent is that if you want to prepare a custom audio mix for the webcast you can attach an external multi channel mixer to those outputs and then send the final mix back into the system using the Webcast in jack. The webcast connection would then receive this custom mix and pass it out to Zoom/Youtube or whatever you are streaming external to the building.
(We don't do this since I can simply take the output of the mixer and feed it into the webcasting computer directly via it's soundcard input)
The computer connects to the Razr Ripsaw or what ever HDMI/Audio capture device you are using using a USB cable that allows the device to emulate a webcam. Using OBS or Zoom on the attached computer you can connect to that webcam input.
Our method to feed audio into the chapel is documented in many of my other posts or at
http://www.federalwaystake.net/Hosting. Fundamentally audio into the chapel from the rack is sent via the HDMI customer panel input and made available on the rostrum HDMI output where it is routed to a projector from which it is extracted from the feed and injected into the sound system using the RCA audio inputs on the rostrum. You could also use the HDMI customer panel input directly to inject audio but we do not do so since that setup is entirely dependent on someone on the stand to turn it on, and if they are untrained this can be a point of failure.
I realize that this is a bit complex and confusing, which is the reason for posting this stuff in the forum to help us all learn and gain a better understanding. Happy to help.