Well, depending on the size of your grow box it could kinda naturally happen through thermal exchanging. But you can never rely on that. You'll need an exchange of fresh air for the plant. The heat build up is going to cause issues as well.
Maybe, just maybe, if you have strategically placed passthroughs. Fresh air intake on the bottom and vents on top for exhaust. Using a slight venturi effect would draw in fresh air when your lights on. But lights off and it'll stagnate. In that case, get some small box fans, 5v computer fans.
I dont think its going to go well. But its worth a shot right? Take pictures of your setup as is. Im sure theres going to be plenty of input from others.
I ran a small fan but no exhaust in my micro grow box I did for fun. Just opening the door couple times a day is enough exchange. I forget the exact dimensions but something like 20x24in and 4ft tall.