They simply file it as an expense when they submit their travel voucher. They're complaining that they were charged but not discussing the fact that they get paid back on the voucher. Every military person traveling goes through the same process.
Those guys returning have been in the desert a long time pulling in salary at tax free and being paid at least $500 extra a month in hazardous fire pay and family separation pay. When you are in the desert, you get paid a but load of money.
Also, they have government travel cards for these expenses. They are not paying out of their pockets for this expense.