First, if you're active duty overseas, why are you buying a car here instead of over there. We've bought a number of cars through military sales in Germany, including one just a few months ago, and gotten fantastic deals.
Anyway, I paid cash for a car a few years ago at Carmax here in DC (Reston), and they didn't require a credit application. I think it was around $22k, and this would have been circa 2004.
I'd walk.
Main reason is I'm looking for a used conversion van - new ones are 3.5 to 4x my budget.
A side note of interest: the economic slow-down has really messed up the delivery time lines for overseas military auto sales. I sat down with one of the salesmen and found only Ford has the assembly line throughput to guarantee a custom build by early June. Was looking for a Ford, but again, well out of my price range.