In my opinion, Oyakodon is more about the technique than the dish, which is X poached in Y liquid and poured over rice. In that spirit, I’ve made many variations of this over the years with beef, chicken, vegetable, and mushroom stock with great success: Dashi is the way of the of the Japanese, but it doesn’t to be your way. 
Now there is a standard Japanese version in which the chicken is swapped out for a whole onion. As an ovo-vegetarian, you could do that.
As for the dashi, you could use a powdered dashi and avoid the fish flakes. TBH, I don’t know how much of the umami in powdered dashi is from an animal or not, but dashi is always about umami and never about a fish taste.
Oh, and for strong dashi, I’m using 30 grams of both kombu and bonito (where the standard dashi is usually around 10 grams each.
Does any of that work for you?
