Do you think that if you swallow a whole lot of air and stored it in your stomach, you would be able to hold your breath longer than if you hadn't?
I think there are two issues to consider here:
1) The regurgitation factor
At first thought, it might be possible to regurgitate some of the oxygen containing air out of your stomach and suck it into your lungs, but considering the fact that there's no way to create the negative pressure in the areas you require it to affect the transfer, I'd say it's impossible to play an active role in getting the oxygen containing air in your stomach to your lungs.
2) The incidental absorption of oxygen
As far as my understanding of pulmonary evolution goes, we wouldn't technically need lungs if we had a much, much larger surface area:number of cells ratio and had our capillaries much closer to the surface of our skin.
Sure, a stomach doesn't have the enormous surface area of your average lung, but surely it's better than our skin... Then again, I imagine there's some kind of mucus layer between the stomach acids and the actual stomach itself.
Maybe if you could somehow fast-track the passing of the air to the upper intestine, that would help. There are all sorts of blood vessels there.