Are there any natural home remedies for a child with social anxiety?

No. There are not proven non-medical remedies (such as homeopathic products, herbal products, foods, or drinks) that are effective for social anxiety in a child. Parents should not feed a child foods that cause symptoms which might increase his anxiety (giving a lactose-intolerant child cheese early during a party can cause stomach aches and gassiness later in the party). Good treatment is behavioral.
Not that I know of. Many think, that if something is "natural, " that means that it is safe. Actually, this is not true. There are many natural substances which are actually harmful. The best thing to do, is to consider having your child evaluated for a problem which could respond to some standard treatments. Such treatments may or may not involve medications. I would start with your child's pediatrician. Good luck.