The Arc doesn’t look for an HDMI signal. It looks for an HDMI ARC (Audio Return Channel) signal, which is normally created by the electronics in a TV set. 

Sonos isn’t really designed for karaoke use, but it’s possible and likely expensive/convoluted that you could get an adapter that strips audio from HDMI and converts it to optical (there are many, and they’re cheap, most HDMI switches do this) and then take an optical cable from that device to the Sonos Audio Adapter that Sonos sells, which takes an optical signal and converts it to HDMI ARC, which you would then plug in to the Sonos Arc.  There are a lot of pieces in that chain, any of which could fail on you. And I’m mildly concerned about the possibility of enough delay in the signal to alter your ability to sing along. As I said earlier, Sonos just hasn’t designed anything for use in this kind of situation. 

Or, you could potentially connect your red and white analog outputs from the karaoke machine to matching inputs on your TV set, and the TV would (hopefully?) send the signal back out through the normal HDMI ARC path to the Sonos Arc. Not 100% sure, but it would be a no cost test. But there’s no guaranty that the TV would necessarily encode the analog input to digital, and send it back out through the ARC port. That may be something you need to check with the TV manufacturer.