Yes, you can use Sunday Mix for a livestream or live performance!
You can totally route audio through Sunday Mix’s audio tracks in real time as long as your computer is up to the task and you have the needed i/o to do so and adjust for latency on the output side. You'll need some sort of broadcast software to get the audio output from Logic synced up with your video.
Some features included in Sunday Mix are only for post production - like the kick and snare replacement. Others like the Sub Kick will work in real time.
Check out this video below to learn how to set up Sunday Mix to record directly into it:
Check out all the video tutorials at the Sunday Mix Resource Page!