Hi, this Livestream Broadcaster is a little box, pre-configured to encode and connect to Livestream.

I don't have that box but a similar device from Teradek (Cube) that also comes with presets for connecting to several telecom providers and streaming providers.
The Livestream Broadcaster looks very similar to the Cube, so they might come from the same suppliers....

I have not used my Cube to encode and stream to Livestream, my first test were just directly connected to a wireless network and then connect to Livestream with the Flash Media server.
That worked reasonable, with a delay of approx. 30 seconds.

Can vMix connect to the broadcaster? sure, if you can output a signal that it can understand. My Cube has SDI input so it must be SDI from a decklink card or such.
The Broadcaster needs HDMI and you can get that out of the video card, normally. The issue might be with the audio which might be handled separately in the computer.
SDI-audio is then more useful because it can be embedded into the stream.

In my system I have besides the Decklink card also an HDMI output from my videocard that includes the audio-stream. The Datavideo HDMI to SDI converter embeds this audio stream in the SDI so they are always locked.

So you need to stream the image (and sound) that you see (and hear) on your HDMI monitor? As I said earlier, I have not worked with the Livestream Broadcaster but it looks to me quite straightforward: connect that same HDMI signal from your computer to the box and you're set. You probably need a HDMI-splitter (that supports audio) to view the monitor AND feed the box.

The Broadcaster will scale and encode the image to make it suitable for the available bandwidth of your internet connection. There is a whole settings menu in that little box too.

Rob