C.H.
I would say....at your DR push for an immuno-cap. It is basically an allergy test by blood. It is really simple. If there are specific things that you would like to test ask your DR, otherwise there is a standard list (milk protein, eggs, and others). If you have any intention of trying almond milk, ask for almonds to be included on the test. I would recommend against just trying it until you know because kids with allergies aren't usually allergic to just one thing & tree nut allergies can be severe.
Regarding your daughter's vomiting. My twins had the exact same thing & sure enough they have a milk protein allergy.
As a side note-- Lactaid won't help as it still contains the milk protein even though it is lactose free. If you don't want to try soy, you can try Rice Dream.
Good luck