Rubber tires are very difficult to re-manufacture into 'new' tires, without waving another Morrow Industries Magic Wand. It'd be easier to just stash a spare tire of some sort in each cache, and thousands more in various depots. Or even just dig a hole, line it with something to keep water and stuff out, fill it with tires, and cover -- not even much of a security risk if discovered. Store enough tires so that by the time you run out of tires, you've probably run out of armored car, too! Have a lot of patch kits on hand.
Various run-flat systems existed for the "Classic" era, and were standard on most of the Cadillac-Gage armored cars: either foam-filling, or solid rubber "doughnuts" mounted inside the regular, pneumatic tire. Neither type are meant to make the tire immune to holes -- rather, they allow you to drive around for some distance.
Oooh, an exciting article about American military tires during WW2:
http://www.usarmymodels.com/ARTICLES/Tires/tires.html
And I think this is the Commando Special tire:
http://www.stausaonline.com/military/commander.html
In any case, notice the deep rubber.
Oh, and there are a couple of big piles of used tires; one is in Colorado, and the other in northern Mexico (near Ciudad Juarez, I think).
If you're running a super-modern campaign, maybe think about airless tires:
http://www.smartplanet.com/blog/bull...-go-flat/14406
http://www.pavementsucks.com/board/t...-Airless-Tires
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Michael B.