Great timing, as this is something I just recently started looking into myself. I just posted a similar thread a few days ago on another knife making forum, and nobody seemed to have any clue what I was talking about. haha
The air lapper you're referring to is actually at 11:30ish. Before that, he's using an oscillating multi-tool. Those you can get an any big box hardware store. I bought a "Tool Shop" brand at Menards several years ago for about $30 IIRC, and it's a great little tool for all kinds of things.
As for the air lapper, jewelers and tool and die/mold makers use something similar for detail polishing. Some times they are referred to as "ultrasonic polishers" and the electric versions can run anywhere from a few hundred to a few thousand dollars. There's some polishers from a Chinese brand labeled under "YCJS" that I've seen on eBay and aliexpress. I'm a little hesitant to tool the trigger on one though, because it's still $400 to $500, and I just don't know how well it would work.
On the the other hand, I did recently find this on on amazon:
which looks like it might be similar to what he was using. The one in the video looks like it holds slightly larger stones though, so I don't know. The video you posted is the first time I've seen anybody using one, as well, but I personally think it's a technique that would really take off if more makers knew about it (as well as what tools and stones to use).