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For spacecraft parts, they absolutely don't mind (they're charging for the privilege of course). For the parts I'm familiar with, we generally buy both the necessary flight-rated components (both enough to build the vehicle and some number of spares) and a number of unrated components used in various test apparatuses in a single order. Once you get down to the level of stuff that's not even a flight-test fixture, we can indeed source parts from pretty much wherever. The biggest issue then become US government procurement rules that require us to buy American, but I'm pretty sure I've seen at least Mouser get used before.