Vermont's 5th congressional district is an obsolete district. It was created in 1821. It was eliminated after the 1840 Census. Its last Congressman was John Mattocks.
Vermont began with district representation when it was admitted as the 14th State in 1791.
From 1813-1821, beginning with the 13th Congress, Vermont elected its US Representatives statewide At-Large
After the 16th Congress, Vermont returned to electing Congressmen from districts
Vermont added the 5th district in 1821, Vermont was apportioned a fifth congressional district after the 1810 Census however this, along with the likewise created sixth district were constituted at-large until 1821, and then from 1823-25.
References
Vermont's 5th congressional district Wikipedia(Text) CC BY-SA