Fetching an order's inventory through the API is simple (it can be done from a browser). Unfortunately, you'll only get a JSON encoded inventory that is not very intelligible to humans. Generally, you would want to convert that into some other specific format, and that requires programming.
BrickSync does the conversion to BrickStore BSX for you. Perhaps there are other softwares or services that can do the same...
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You seem to synchronize inventories. If you use BrickSync, then all received orders are saved as BrickStore/BrickStock BSX files, in data/orders/
BrickSync does the conversion to BrickStore BSX for you. Perhaps there are other softwares or services that can do the same...