Is an audio interface device needed to stream to FB or YT?
No
But I highly doubt that is your actual question
I'm suspecting something more like - Do you need to add an Audio Source to OBS? and the answer is.. it depends. ONe of your other sources could include Audio, so no, you don't need a separate audio source.
BUT
if your existing OBS sources do NOT include the audio you want (ie your mixer), then yes of course you need to add that to OBS as a separate, to-be-composited source. This is common when you have a camera that does not include audio, or you don't want to use the audio from a webcam, DSL, PTZ camera, or whatever, and instead use (presumably) better audio from an alternate source (ie mixer).
but it depends on each person unique setup, gear, connections, etc. For example, your mixer output could be routed into your camera so audio sync be handle as single HDMI source... just depends
