Background

Internationally adopted children tend to suffer more from micronutrient (vitamin and mineral) deficiencies and lack of attuned, stimulating care than from protein/calorie "macronutrient" malnutrition. Basically, most kids in orphanages get enough protein and calories to grow, but don't because of stress, neglect, and, perhaps, some micronutrient deficiencies. Iron and iodine deficiencies are well documented, as are vitamin A and D deficiencies.

The iron deficiencies (+/- anemia) may worsen during the catchup growth period, as children outstrip their limited iron supplies. This makes getting extra iron important for most internationally adopted children, for as long as they're having catchup growth. The constipation angle is overrated, in my experience, and I'd rather manage the constipation than see your child's brain development impaired by lack of adequate iron.

Vitamin and trace mineral deficiencies may also be implicated in the high rate of initial skin and hair symptoms. Zinc deficiency has been linked to stunting, poor healing, diarrhea, and cognitive delays in developing countries. I also wonder about essential fatty acid levels, and if we should be doing something to support the rapid brain growth we so often see. Many of the placing countries in international adoption also have environmental toxin issues (former Soviet Union, India, and China among others), but fortunately the lead issues we saw in years past are better these days.

In general, the research supports prevention and correction of deficiency, but not so much the trendy "giving more of a good thing" supplements industry for children not at risk for deficiency. A varied diet seems better for children at low risk of deficiency, although we still do recommend 600IU per day of supplemental Vitamin D for kids (Ddrops, Carlson D Drops, etc).

Vitamins and Minerals

While vitamins and minerals are best absorbed from healthy food sources, it can be hard to meet the complete nutritional needs of a rapidly growing adopted child (who probably arrived with micronutrient deficiencies) through diet alone, especially if they're picky or have oral-motor delays and sensitivities. For that reason, we recommend as complete a multivitamin and multimineral supplement as you can find, at least for the initial 3-6 months of catchup growth.

The ideal supplement for internationally adopted children would contain plenty (100% RDA) of Vitamin's A and D, iodine, iron, zinc, selenium, and other vitamins and trace minerals. I've not yet found the ideal liquid/powder form, and would love to see suggestions in the comments below if you find a good one. Plenty of options in the chewable format, which can be crushed, but that's extra work, and not so dissolvable.

Enfamil's Poly-Vi-Sol with iron is a basic multivitamin plus iron liquid supplement that is easy to find, but not as complete as the others.
Novaferrum makes a multi with iron liquid on Amazon that many families feel taste better than Poly-Vi-Sol.
As for chewables, there are lots of options out there. Flintstone Complete and other drugstore "complete" multivitamins are easy to find, and taste good. Heck, they even make vitamin pixie sticks these days.
Remember that iron supplements are a frequent cause of accidental overdose in children - the doses we recommend are quite safe, but please keep the tasty vitamins well out-of-reach.

Essential Fatty Acids

We also sometimes recommend essential fatty acid supplements like fish oils for new arrivals, as it seems plausible that children from malnourished pregnancies, who weren't breastfed, who were raised on diets poor in healthy omega-fatty-acids, and who are having rapid brain growth may benefit from supplementation in this area.

Essential-fatty-acid (EFA) supplementation has shown decidedly mixed results in ADHD, which is more common in IA children. DHA is a type of omega-3 fatty acid that seems important in early brain and vision development, and is a major structural building block in the brain. EPA is another omega-3 fatty acid that may be more helpful in later issues like attention (ADHD) and mental health. Flaxseed oil contains ALA, some of which is converted to DHA/EPA.

The optimum ratios of DHA and EPA have not been fully worked out, but I like to see more DHA early on for infants and toddlers, and more EPA for older children. Some fish oil products even include some healthy omega-6 and omega-9 fatty acids, for balance. Cod liver oil is a grandma favorite that usually contains natural vitamins A & D - check the labels to make sure you're not overdoing these vitamins, especially if they're in your other supplements and formula. It would be nice to get these healthy fats from diet alone, but sadly, our fish supply isn't safe enough in terms of mercury and PCBs to safely consume enough to meet our target intakes of DHA and EPA.

Some quality fish oils that are independently tested to have adequate potency and very low levels of contaminants include:

Nordic Naturals - lots of options here, including flavored oils and small chewable gelcaps
Pharmax's Finest Pure Fish Oil is one of the less fishy oils out there, and their more expensive Frutol is a fish oil that's emulsified with prebiotics and fruit purees. They even make powdered versions, but I hear those are a bit fishier. Available where their probiotics are found.
Coromega is another pricey emulsion in orange and chocolate pudding flavors for kids that won't tolerate fish oils straight up, cheaper through VitaCost.
Carlson's Fish Oils are also easy to find, and available in child-friendly formats.
Costco Kirkland Brand fish oil softgels are inexpensive option for folks that can swallow pills.
Tips for taking fish oils - you can often get kids used to taking the oil straight, or try them stirred into a "shot" of juice or water, smoothies, or applesauce.
Flaxseed oil is a vegetarian option for omega-3's. Refrigerate these oils, as they go rancid pretty easily. Ground flaxseed sprinkled on food or in baked goods is another way to go. While the ALA in flaxseed may be important in its own right, it's not very predictably converted to DHA and EPA.

Probiotics

Another potentially helpful supplement would be probiotics, which are the good bacteria that live in your digestive tract, and that are found in yogurt. In fact, there's about 3 pounds of these bacteria in your body right now - isn't that a lovely thought? The Europeans have been big fans of probiotics for a while, and pediatricians are just starting to catch on. Probiotics seem to be a safe thing to try, especially for children with loose stools or those taking antibiotics. Since children from orphanages (where antibiotics are overused) are likely to have less-healthy "institutional strains" of these gut bacteria, it may be a reasonable thing to supplement for IA children. We've got lots more info and recommended formulations in our "Probiotics and Prebiotics" article.