omg yes of course the dyes are safe for dogs!!! They're all vegetable based dyes. There are human ones (manic panic brand; tested on people, not animals!) but since the colouring craze has hit around the world, there are specifically made pet dyes now too!
The orange is an orange dye... You just gotta find one that has a nice orange; the grooming forums around the place would know which ones come out best.