Yeah, not surprising that the guide isn't clear on this, it's not clear on a lot of things (I also loved the vast supply of resources offered for things like vehicle stunt rings... "Either do it alone or here's how to do with buddies to make it super easy." Yeah thanks for all the details on doing it by yourself such as things like links to videos, much appreciated guide author. That's okay, don't make a detailed guide, I'll just go look for the information myself, wouldn't want you to exert yourself.
My guess would be that since the achievement specifically says "Win a Vehicle Tag Quick Match..." that it won't work in Private as it tells you to do it in Quick Matches. You can still boost though if that's what you want, you just have to get 8 friends together and kind of take over a room: as people playing leave, get your friends to join up quick. Eventually you'll be able to get all 8 of your buddies in and then the boosting can start.