I've tried to replicate the drink at home, but it is never quite as tasty as the shop down the street. Not sure why (although, I don't do the tapioca at home, too labor intensive) They also have a Thai Iced Tea version, which is also yummy but very sweet. I don't like it too sweet and have asked them to make mine with less sugar.
I don't see why you couldn't make it with the vanilla soy milk, I know they offer soy as an alternative (but it's regular, not vanilla). I've had a version with almond milk, too.
Here's a visual:
http://tinyurl.com/3e3nktl