Ok, 2 symptoms, throwing up and scrawnyness. My head's still swimming from going through the books, but didn't really find anything about throwing up. Since that's just not something chickens do, I'd think that it would appear as a symptom of any disease or parasite infestation where it occured. Scrawnyness, well, if she's always been like that, not really a symptom. So, we haven't gained any ground. The other hens are apparently fine, therefore we can assume it isn't something spreadable.(posted 9670 days ago)In all honesty, I think she's just a runt, who is surviving only because you and the children are caring for her. She may be getting her tail kicked by the other hens when humans aren't looking, but usually those bullied birds show other signs including lack of feathers from picking attacks.
I can give you some diseases and parasites that very generously could vaguely cover your hen, but I don't think it is worth the typing. They're pretty much fatal anyway. Since she's survived almost a year, and while there's a remote chance she's a carrier, I really don't think any of them apply.
You could, if the kids have a good microscope, get a stool sample and check it for anything odd looking. Or take the sample to the vet. It probably isn't worth sending it off for serious testing, again so much of this stuff isn't treatable it wouldn't be worth it.
I'd make sure she's getting plenty of real chicken grit, vitamin A (greens and yellow corn, squash, etc) possibly sprinkle some vitamin B powder on her food (start with a little and increase if she doesn't have a problem with it). You might want to make extra sure she gets treats and say cooked grains. It will help her a lot if you can get some weight back on her.
If the kids continue to help her, she may live for some time yet, but I'm afraid I'd prepare them for the possibility that she isn't now and never has been a healthy hen. Gerbil