Hi,
Probably one of the most important supplements, especially with any autoimmune disease, particularly in the cold, and as the days are getting shorter, would be Vitamin D.
Current scientific research suggests that all the cells and tissues in your body have vitamin D receptors and require vitamin D for their well-being. Vitamin D is also responsible for the regulation of over 2,000 genes in your body. It's crucial to just about every complex metabolic process in your body.
I'd say that if you can only get them to take one supplement, this will make the greatest impact on their health (And, I am not really a fan at all of supplements, preferring to simply offer as many whole, organic foods as possible. I don't know if you're familiar with Dr. Mercola or not, but he has compiled a lot of good information on all aspects of Vitamin D. here's just one of his articles, though there are many:
products.mercola.com/vitamin-d-spray/
I've got six kids myself (my youngest are twins-they were triplets, but I miscarried their triplet at 10 weeks), so well familiar with the 'picky eater' syndrome. To make it easier on me, I just tried to get away with serving as many simply or minimally prepared foods as possible. By that, I mean something that didn't have to be cooked, really, but rather just washed and arranged decoratively on plates(we are vegetarian).
One thing that I did, and even now, as my kids are older, they remember and love, was I picked up all kinds of pretty little plates, saucers, teacups and teapots at thrift stores and garage sales. Also, I bought pretty little vases, so that the kids could go out and pick some fresh flowers each day for the table. Rather than lunch, we'd have 'high tea'--a colorful array of fresh fruits and veggies, nuts and seeds or little tea sandwiches. It always seemd they would eat more, or try new things, when served in these small quantities on the decorative plates. Oh, and kids that age love to dip their stuff, so hummus or a yogurt/dill dip or guacamole are good. I had a couple of Tea Party books that had some fun, simple recipes, and my girls especially loved to make up entire 'themed' tea parties. But, there's just something about making the meal festive, and having the kids do as much as possible in preparing the food, as well as the table, that even the pickiest one (and EVERY family has one!) would manage to eat something good for them. That way, too, I didn't feel we had to totally cut out sweets (which was hard for me to do), because they would be tiny portions, just a little dainty sweet now and then, and almost always homemade, so I knew what was in it.
You might also want to look into homeopathic care for your kids. (I'm not referring to supplements or vitamins, but to actual homeopathic medicine.) It's very effective for autoimmune problems. It takes the entire imbalance of the body into account, including the genetic weaknesses that all of us come into the world with. Our bodies have an amazing capacity to heal themselves.
Blessings to you and your kids,
April