There are 16 US Post Offices in Addison County serving a total of 35,029 residents equating to 2,189 residents per post office. It's estimated that approximately 55,910 packages are handled across the 16 county post offices annually. Addison County is a county located in the state of Vermont, in the United States. At the 2010 census, the population was 36,821. Its shire town (county seat) is Middlebury."