Since we keep Lugol's in tightly capped amber glass, I would only dispense Lugol's right before using. You may already know, but adding Lugol's to orange juice negates the elemental iodine for which many people take Lugol's. Your kids are getting only iodide in their juice.
The problem isn't the orange juice acid, so mixing Lugol's with cider vinegar isn't a problem. The problem is the ascorbic acid of the orange juice, as well as the vitamin C crystals you're adding to it. In fact, when someone has spilled Lugol's and gotten the iodine stain (from the elemental part), one of the recommendations is to use ascorbic acid (vitamin C) crystals to neutralize the stain. You can test this by adding 5 drops or so of Lugol's to a couple ounces of water, enough to see the brownish color, then add ascorbic acid, stir, and watch the water clear.