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I have been searching, unsuccessfully, for a way to view BBS entries across all stations in a sector without having to dock at each one. Is such a script possible, similar to how you can see the trade goods and prices of any given station without docking?

This is not possible to do. Scripts can only add entries to the BBs, they cant check whats already on them. Also, the articles are generated when you initial view the BBS, so if you havn't viewed them, there will be nothing available