Ok, truthfully, I'm not a fan of electrical, but since this was up all day and no one has wanted to help you I'll take a stab at it for what it's worth lol...
"Wondering if anyone has installed this or something similar on their Vespa?"
No but I've helped install, on KYMCO scooters, wireless third brake lights. All you needed to be able to do with that OEM kit was install a tail light bulb, and then stick the 3rd brake light to the back of a top case or possibly a helmet.
I kind of wish GoGoGear had done a similar wireless system where you just carry a little battery in your pocket, as opposed to cutting into the harness.
As a general note on cutting into your wiring harness, does this jacket include any kind of connectors? If not you can get some easy-peasy butt connector and crimper from Harbor Freight for $10:
https://www.harborfreight.com/electrical/electrician-s-tools/plugs-terminals-connectors/305-piece-solderless-terminal-kit-67684.html
Similar sets on Amazon seem to go for 3x the price based on a quick look.
Some people, maybe even smarter ones than me, might suggest getting a cheap soldering iron, rather than crimping. If it's your first time getting into your wiring harness you might just want to go the easy root with the $10 kit.
Sorry for the very general answer. If you have a more specific question I can try again.