The best method is the “elimination diet.” Your child goes on a strict diet, eating only what you know does not cause allergic reactions. Every week, you add one new food, observing for any symptoms.
If you notice that he’s allergic to it, you remove it and then revert to the previous diet for three days. Then, you introduce another new food. This goes on for several weeks or months, until you finally have a complete list of what your child can or can’t eat. Always do this under the supervision of a doctor.