My family tried Yak and Yeti this past trip...My brother is deathly allergic to tree nuts and is a celiac. He was able to order a few options but the safest was the steak and chicken combo (usually with tempura shrimp). The chef was helpful in pointing out what he could and could not eat... Unfortunately they are not able to accommodate as easily as some Disney owned restaurants (they are owned by the Landry Comp. same as Rainforest Cafe, T-Rex Cafe, and Bubba Gumps).
I am not sure about your sons other allergies but hopefully someone else on the boards can! I do know though that as far as the tree nuts and gluten your son will be in good hands in Disney they take every precaution and made my families stay wonderful!
Your options may be a lot more limited. Unlike at the WDW restaurants that can whip up special items, Yak and Yeti seemed to only be able to make some modifications to what's on the menu. We ate there two trips ago but then skipped it last year because there just isn't enough I can eat.
I have egg,chicken.fish,shellfish,banana,kiwi,and avocado allergies and ate there this past May it was a little challenging but was doable I had the ribs with mashed potatoes which he wouldnt be able to get since they have milk in them