Is there a method to record video chat app (google duo, skype, tango, etc) straight to the phone?
I'd recommend you to use Az Screen Recorder. 


  I'm of the belief the device you want to run this on is a relatively
  new device supporting Android Marshmallow. That's cause Android added
  that package as an extra feature for 5.0 devices and above.


Az screen recorder will record video without issue. When it comes to audio it uses the mic to stream audio. So you'd need to make your call on speakers and not using headphones or Bluetooth speakers. I personally haven't tried this and will update my answer once I do. But with my previous experience using the app it should work.