The Mic/Guitar input can be recorded as audio using the USB Audio function on both the PSR-S970 and the S770, however the S770 does not have a built in Vocal Harmoniser. The USB Audio recording function on both keyboards only records to a stereo track and does not allow 'multi tracking' or overdubbing.
So using either the S970 or the S770, you could record the MIDI parts from the keyboard in up to 16 tracks/channels of the MIDI Song Player/Recorder, editing or re-recording them as necessary until you are happy with the song.
Then set the levels of the MIDI song and the Mic or Guitar to balance each other.
Finally start the USB Audio recorder, then start the Song Player (which will produce the audio for the recording), and play the guitar or sing along with the MIDI playback to produce a stereo '.wav' audio file.
If this method works for you then fine. However one problem with doing this is that you can't readily edit the resulting audio, either on the keyboard or even on a computer (if everything is on a single stereo pair of tracks), so if you make a mistake when singing or playing the guitar, you really have to redo the whole take again from the start.
Also because you can't overdub the audio file using the keyboard, if you wanted to play guitar and sing, you would need to do both at the same time meaning that you would need to connect the Mic and guitar to a small mixer, and connect one of it's outputs to the single Mic/Guitar input on the keyboard.
A better way might be to record just the 'backing' MIDI parts of the song to USB audio as above, but then transfer the '.wav' file to a computer DAW (such as the free 'Audacity') and add the guitar and vocal parts to two new tracks in the DAW. Ideally to do this you would need an audio card in your computer or an external USB Audio Interface to get the vocal and guitar parts into the computer.
However there may be a way that you can use the keyboard to record the vocal and guitar on separate audio tracks without requiring a mixer or external USB audio interface, but still requiring a computer DAW.
It should be possible to record the vocal and guitar parts as individual '.wav' files using the keyboard's USB Audio recorder. To do this you would need to import and monitor the playback of the 'backing' track from the computer DAW through headphones and sing along in time with this. If you want to play guitar as well, then make another recording with it in the same way.
An advantage of this approach of recording the vocal and guitar parts separately, is that you can apply the most appropriate effects for the particular part being recorded using the single DSP4 effects block that is available for the signal from the keyboard's Mic/Guitar input.
You would then have one or two more stereo '.wav' files which you could import into separate tracks in the DAW. Most DAWs should allow you to move each complete track timewise, so you can 'line up' or synchronise all tracks correctly. If you are not happy with either of the new parts, you can just repeat that one without affecting the others.
Once you are happy you can mix down all tracks to a single stereo pair which can be saved as a '.wav' or 'mp3' file and played back on other equipment.