I don't think you will be able to find a full size (27' wide) washer and dryer that will fit that depth. From my experience, having dealt with a fairly shallow laundry closet, only the LGs are under 30 inches. However, you need space for a dryer duct and hoses,etc. Even using a rigid flat duct, you need at least 34-35' depth. So I think your options are 24' Euro style machines (that would be my choice if I were you) or a small stacked unit.

What if i purchased a side air vent since the exterior vent is on the left side wall?

That would probably help but you still need space for the hoses and the drain pipe from the washer (at least 2-3'). So 31' will super tight. Are you measuring to the wall/trim or the door? The door frame may give you a bit of additional depth.

Do some research and look at owner's manuals for various brands and see what you can find. They give dimensions as well minimum clearance required from the wall. I can tell you right now that any Samsung model is out - they all are over 30'. But you may have some luck with LGs, Duets or other brands.

Good luck.