Hi, M.. Rice milk is a great substitute for the comfort and convenience of milk. Unfortunately, it's not nearly as nutritious as cow's milk or soy milk, which your little one can't have. So, yes, I would give her some supplements.
Rice milk has only a modest amount of protein, vitamins, etc. If it's made from whole grain rice, it does have some iron in it. Mostly, it's a great thing to drink, and it's great for cereal and to substitute for other uses of milk, but most brands of it just don't have a lot of good stuff in them.
Try to balance out things with more healthy proteins. To supplement some of the minerals, try vegetable juices or smoothies with great combinations of stuff that she's not allergic to. Can she eat lactose-free cheese? That's a pretty good source of protein, calcium and magnesium.
There are some wonderful allergy cookbooks on the market today. I'm sorry, but I've forgotten their titles. You can google "allergy cookbook," and get a whole list.
I wish for your whole family happiness and health.
Peace,
syl