The government of the Shire, insofar as it had such a thing at all, was extremely loosely organised. There was only a single 'official', the Mayor of Michel Delving, who was responsible for day-to-day matters such as running the Messenger Service and overseeing the Shirriffs. There also existed the Shire-moot, of which we know nothing for sure except that the Thain was traditionally its master.
If the moot of the Shire followed the real historical pattern from which it takes its name, then it would be a meeting of Hobbits from across the Shire to hold a court, or otherwise discuss matters needing attention. A Shire-moot would typically be attended by representatives from the most important towns within a region, and the arrangement of Folklands within the Shire would mean that in practice the senior members of the most important Hobbit families would be in attendance when the Shire-moot met.
Apart from the Thain, we're not told exactly who would have attended meetings of the moot. Bilbo Baggins is one possible candidate: as head of the Baggins family he would have been the natural representative for his home town of Hobbiton, but no record exists of his ever being called to attend a moot.
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