Capturing the incoming voice chat means, capturing "desktop audio".
What you would need is a route, either with a audio cable or digital via software, to route that audio into your capture card/stream.
But remember, that this might result in loopback.
Question as always is: How and with what gears do you do what you do. A technical drawing would light up the way.
The simple streaming setup PDF shows a basic scenario, which is independent from gears and software.
Please consider to create such a technical drawing for your scenario along with the gears and software in charge.