Mr. Bingley wanted news of Miss Bennet early in the morning. It's quite possible that a housemaid brought tea in the morning, stoked a fire, opened some shutters, etc. not very long after he arose. Neither his housekeeper (who would not be doing such chores and who would be busy anyway) nor his sisters' maids (who would most likely not have arisen yet themselves, being "elegant") would have been accessible early in the morning. Since he must send a woman to inquire about a female guest, the housemaid was there and the only sensible choice.