At our church we do two services and produce record both with vMix 4k but only live stream the second service with vMix. Sometimes it would be nice to just stream the recording of the first service but we still want to record the second service with vMix. Is there another program (preferably free or inexpensive) that I can stream the first service recording outside of and independently from vMix that won't interfere with our recording the second service? Seems kind of odd maybe but sometimes the pastor really liked the first service sermon and wants to take the pressure off from knowing the second service is live.

Thanks in advance for any advice!

You can do all that in one copy of vMix..

set the input that is the previously recorded video to Output 2 and put that on a separate audio bus
Set the 2nd stream key to a separate stream, and set that to output to output 2 with the same audio source as the video

That will have the recorded video streaming on its own

then just run your output 1 stream like normal to a different stream key and that's it. You can record the second show at the same time as well.