Proper cook times for mushrooms, onions, and peppers
I am going to make the assumption that you're stir-frying in a really hot pan. The onions and peppers are going to take around 5-8 minutes to get good color but retain their snap.

I would cook the mushrooms separately - they're going to take a little longer 8-10 minutes, because they give off a bit of moisture.

So as you're poaching the chicken I would 


Fry the onions/peppers - remove and keep warm
Fry Mushrooms
When the mushrooms are done, add everything else to pan to meld flavors.