Some of the non-vegan foods are among the most causes of allergies: shellfish, eggs, and dairy. How are these tests prepared? Is it possible for the allergens to be extracted without harming the animals they come from?
I believe it's impossible.
To test for allergies, you must find a way to expose your immune system to the allergen. This can be done by applying the allergen directly to your body (as in skin prick tests), which would require a sample of the allergen from the animal.
I'm not sure @Ramon Melo is right that human lactose would necessarily cause a reaction in people that are intolerant to cow lactose, because lactose differs slightly between different species. For example, many lactose-intolerant people can tolerate goat's milk just fine.
It could be possible to isolate lactose from one animal, and then bioengineer yeast cells to produce it. While this would require animal ingredients initially, the yeast can produce the allergen without any additional animal ingredients. And since yeast is a fungus, I think most vegans will be fine with it :) But again, it does need animal ingredients initially.
Alternatively, you could do a test to search for antibodies in your blood that will react against that allergen. But these kinds of tests (e.g. ELIZA) use linking antibodies that are produced in animals like chicks, rabbits, etc. Very far from being vegan.
Sorry, folks.... that's a bummer!