The Army & Navy stores were originally a co-operative to supply kit to members of the armed forces as they headed off around the world to guard the British empire.
https://en.wikipedia.org/wiki/Army_%...ted_Kingdom%29
That box you have there is unlikely to be specifically for a Webley air gun, it looks like it would be for a shotgun or a double barreled sporting rifle, but if a Webley airgun fitted inside I suppose it could be used for one. The chap who owned it originally could have been posted to Burma, India, Africa & all points between. He would have bought it specifically to protect his prize fire arm against the rigours of the journey & the harsh conditions during his tour abroad. This would have been for his personal property, not the guns he would be issued with on duty.
There are quite a few around in the antiques market, but your's isn't particularly good condition, like this one: http://ajbells.co.uk/accessories-antique-gun-case so a guess at valuation would be £30 - 50. It's worth noting though stuff like this can get ridiculous prices from Boutique antique dealers supplying stuff to shops & bars etc. to dress the place up.
This guy does that, so give him an e-mail & see what he says: http://www.drewpritchard.co.uk/pages/contact
I quite like his TV show, he seems a decent bloke, & assuming he gets back to you he should have firm information, a lot more practical than mine anyway.
I think it might be a bit older than the 1930's, you could to do a bit of research on the addresses on the label to find out definitively. I don't think that would do much for the value but it would be interesting to find out.