Just to clarify, our whip cream is made of three things: 1) Heavy Whipping Cream from whatever your local dairy is, 2) Our Vanilla Syrup (which is also in the hot chocolate) and 3) CO2.
I am a barista and I make whip creams every day I work, so I hope this helps.
Also, the only way I can see cross contamination ever happening is in the cold beverage area at the blender. Since we simply just rinse our blenders after each beverage we make, if your drink is made directly after a Vanilla Bean Frappuccino or a Cafe Vanilla Frappuccino, I can see little tiny particles of that powder hanging on underneath the blade of the blender. At my store, hardly anyone asks for anything that is NOT Dairy or Soy, so you may be able to get away with asking for them to use the "Yellow Non-Dairy Blender" and just have them wash it right afterward in the dishwasher (actually a sanitizer, but you get the point) if you want something WITH dairy or soy blended. At least, that would be the most practical idea at my store.
This might not work at Starbucks inside of Targets, Vons, Smiths, Airports, hotels, etc... as those aren't actually owned by Starbucks. They just license the place they are inside of to use their name.