You should hop on over the disABILITIES! board and take a look around. You'll see plenty of reviews by people with allergies including but not limitted to milk.
Every TS and CS restaurant and food court we've been to around WDW has carried Silk soy milk (at food courts it's in the self-serve cooler). Our last trip they seemed to have plain, very vanilla and chocolate. The food courts also seem to all have rice milk (Pacific brand) but you have to ask a chef to get you a cup as they don't carry single serving cartons. At TS restaurants most seem to carry rice milk but I haven't asked at CS restaurants in parks but I get the impression from the Special Diets department that it's much less stocked at those locations. No almond milk anywhere.
I generally place a grocery order with Garden Grocery and include milk as part of the order (I have to order 3 different kinds since 1 kid drinks Silk, 1 kid can't tollerate silk so drinks rice milk and DH drinks lactose free milk; I can drink both the rice and soy). I also order cases of bottled water, snacks and other breakfast foods for eating our room. I CAN get breakfast foods at resort food courts despite our long lists of allergies but I prefer to eat in our room as it's so much faster not to mention cheaper. We eat big early lunches and like our light easy breakfasts in our room.
I absolutely agree that baked goods and breakfast foods like pancakes are wonderful with apple juice. I actually learned that one from a chef at POFQ.
Just so you know, the regular breads around WDW do contain milk with the exception of the gourmet hot dog buns (this is new so I haven't tried them for my daughter yet but I had a chef confirm this for me less than 2 weeks ago). Other than that one kind of bun, the only milk free breads are also gluten free. The regular chicken tenders (kids nuggets and adult tenders) also contain milk but the allergy free tenders at DELICIOUS and completely milk free.
Also, there are lots of places around WDW where you can get Tofutti (soy) or Rice Dream (rice) ice creams. YUMMY!! Most TS locations and resort food courts and select park CS locations carry then as do the ice cream parlours. The popcorn and soft salted pretzels at the carts around the parks are also milk free (not sure about the pretzels in Germany but the standard salted ones everywhere else are milk free) as are the frozen pineapple bars, frozen strawberry bars and Itzkadoozie popsicles at all the frozen snack carts around the parks.