IIRC, part of the problem is long exposure to heat. If you boil the veggies, they will spend a lot of time in hot water, which will not so much "wash out" the vitamins but rather tear up their chemical structure. (This is less likely to occur in steaming or stir-frying because while the heat is even more intense, the exposure time is a lot shorter.)
So no, they will not become part of the rice - not all of them, anyway. They'll have decomposed.

But shouldn't you also be abe to get that orange colour by adding finely grated carrots towards the end of the cooking? It's a pretty intense colourant after all.