Flax seed is a legume and some kids who are allergic to peanuts, soy, beans, or other legumes are also allergic to flax seed. My son is allergic to both soy and peanuts, but, thankfully, never had any problems with flax seed.
It's probably a coincidence that the veggies your son is reacting to are the same color; but, the suggestion about organic vs. non-organic produce is a good one. There was an article recently about the wax used on non-organic fruits and veggies. Seems the wax can contain soy, corn or casein. Scary. Doesn't help you with the carrots or sweet potato (they don't wax those. I don't think) but it does make you wonder...also, if you suspect a problem with latex, sometimes the employees wear latex gloves to stock.