I setup voice.ai, tried recording a sound byte in Vegas Pro and it just worked without any extra steps whatsoever. The only thing I did was open Vegas Pro after I’ve opened voice.ai and set everything up. My settings were the same as in your video – Live Mode [on], Voice Changer [on], Hear Yourself [off].
Then I selected an Audio Track and started recording: Ctrl + R or click the red circle at the bottom left of the timeline (also found under the video preview, but there it might be hidden by default).
I could also toggle the voice changer and it would toggle in the recording as well.
The only potential problem I noticed was that comparing my base microphone recording volume to the voice.ai live – the voice changer is significantly more quiet. Though that’s likely down to some settings as I just left everything default + you can normalize the audio if you wanted to.
So it should just work as long as you could just record your default mic in Vegas Pro beforehand.