Room service can supply juice when the restaurants are closed. They have apple, orange, grapefruit, tomato, V-8, cranberry, and probably some that I'm forgetting.
You can get soy milk at the restaurants. Your server will be happy to fix a "to go" cup for you which you can store in your stateroom refrigerator so your child can have it first thing in the morning or after nap time. It is available in all of the restaurants including Cabanas during the hours they are open. If you don't see it at Cabanas, ask for it. It will be helpful if you have your TA note on your reservation that your child needs this product. It is stocked on the ship for each cruise, but if they know of needs, they will increase the quantity stocked.
If you are on the Magic or Wonder, you'll have a cooling box in your room rather than a real refrigerator. Place a cup of ice in there along with whatever other items you want to chill (like the milk) and it will stay wonderfully cold.