Clancy was born in Detroit, Michigan where he attended the public schools. He graduated from the literary department of the University of Michigan at Ann Arbor in 1907 and he later studied law there for one year. He worked as a reporter on Detroit newspapers for four years before serving as secretary to Congressman Frank E. Doremus from 1911 to 1913. He then served as secretary to Assistant United States Secretary of Commerce Edwin F. Sweet from 1913 to 1917. During World War I, he was manager of the War Trade Board at Detroit, chief inspector of purchases in Michigan for the Medical Corps of the War Department, and recruiting officer of the aviation division in Detroit. He was United States customs appraiser for Michigan from 1917 to 1922.

In 1922, Clancy was elected as a Democrat from Michigan's 1st congressional district to the Sixty-eighth Congress, from March 4, 1923 to March 3, 1925. He was defeated by Republican John B. Sosnowski in the 1924 election. After leaving Congress, he engaged in the real-estate business until the next election. In the 1926 election, he switched parties and ran as a Republican, defeating the incumbent Sosnowski in the primary, and going on to defeat the Democrat candidate William M. Donnelly in the general election for a seat in the Seventieth Congress. In 1928 and 1930, Clancy again defeated Sosnowski in the Republican primary and Donnelly in the general election to be re-elected to the Seventy-first and Seventy-second Congresses, serving from March 4, 1927 to March 3, 1933.

In 1932, Clancy was a candidate in the Fourteenth Congressional District in Michigan, due to redistricting after the 1930 Census. Clancy lost to Democrat Carl M. Weideman, after which he was engaged in an executive capacity with a manufacturing company until his retirement in 1948. He died in Detroit and is interred there in Mount Olivet Cemetery.