I do believe that freeze-dryers have to have a condenser that actively removes the water vapor. It would depend on the size of the vessel, but the water vapor from the food is likely to be enough to raise the pressure to near atmospheric, and certainly enough to destroy the low pressure required. So a condenser is actively removing the moisture by exposing a surface at a much lower temperature than the atmosphere. But come to think of it, I'm not quite sure how this works considering that the condensed water would be ice.
Directly to your question, I am fairly positive it fills the entire vessel. It is, after all, a gas. There is little in the way of other competing gases and there are negligible effects from things like gravity. I would imagine that the gas would be a near homogeneous mixture of the water vapor and other sparse gases.