As I understand it, you don't -need- ANY gas flow from the torch, if your chamber is purged and sealed well enough.  If its just slightly leaky, you could use your torch flow to maintain inflation.  If its sealed against pressure, your torch flow would just over-inflate it, and is not helpful.
In fact, you should be able to successfully TIG weld with a tungsten clamped into a stick stinger handle (if you're careful enough not to let an arc touch anything other than the tip)!  No need for a gas lens, or even an alumina cup.  No worries about stickout, etc.

BUT, such a chamber is limited to small parts, and needs to be purged itself, so I wouldn't expect gas savings.
Sure, it would work for MIG, but you need to make sure you have no gas leaks back your liner, and once again, you're limited to welding parts that -fit- inside the box, so with limited amount of welding, the advantages of MIG are less apparent.

Such chambers are commercially available.  However, Most GMA welding is done with an active gas, so it is not practical for that application, unless the active gas component is Helium.  At the lower and middle amperages gas flow is not needed for cooling GTA air cooled torches.  I still used a flow rate of 10 CFH with a # 6 Gas Diffuser lens

i cannot remember the manufacturers name, but the one I had experience with, had a 1 1/2 inch thick Aluminum floor plate, where all the gas and electrical penetrations were fitted.  The top was thick flexible clear plastic with four glove fittings.   Leather gloves were fitted over the rubber glove box gloves to protect them.  It was about 4 ft. in diameter, so it took a lot of Argon to purge.  You had to cover the pieces wit a guard while sucking a vacuum.  This was to protect the bladder against puncture during vacuum suck down and from melting after welding.  In practice several piece to be welded were loaded at one time.  I understand that there are several variations on the bladder construction, including a variant with a 3/8 inch thick solid Lexan top.  I know one man who made his own glove boxes with plywood and plexiglass.  

warning: Welding engineer / wannabe know it all :)

generally inert chambers are use for titanium, zirconium or other exotic alloys that are extremely prone to contamination and can't handle oxygen in the PPM range. Stainless steel can usually get sufficient shielding from the gun itself or a trailing blanket.

I don't believe their would be any gas savings unless it's a part with an extremely large amount of welds, as the volume of of shielding gas going through the gun is relatively small. Also for most chambers you have to purge out all your shielding gas everytime you load a new part in or out, that in and of itself would consume a large quantity of gas.

air cooled torches generally rely on the shielding gas for some help with cooling, but this is not a concern if you are far below the max amperage and thus not operating near the rated duty cycle.

I've never heard of this with GMAW welding as  that isn't generally considered a "precise" process and creates a considerable amount of spatter that I would assume might have issues with melting the chamber walls. In addition generally GMAW is done with an active gas or inert active gas combo thus defeating the purpose of an interting chamber.

Welding in an inert chamber is generally much slower/ uneconomical in a production setting unless required due to the issues of contamination.