No. You'd need to do it the other way around.
Something like DXtory can do this I believe. You'd cap the game with DXtory and configure it to record to disk, then set up its virtual camera interface as a video capture device in OBS, and stream out from there. This will also cause a higher system load as you'll effectively be double-encoding video... once to record the gameplay-only to disk, and once for the stream.
Be aware that if you use any kind of follower alert noise or music, DXtory will record all of that too in most cases. You may be able to set up VAC or VB-Audio to reroute the sound, but that can get in-depth quickly. Google is your friend.