The newly opened database for The American Family Immigration History Center at Ellis Island www.ellisislandrecords.orghas 162 entries for TARPEY, 44 entries for TARPY, and 72 entries for TORPEY.
This database covers arrivals from January 1892 until December 1924. Once you find something that looks interesting, make sure you view the original ship's manifest, which has more information than that included in the index.
The site is hard to access right now because demand is so heavy, but keep trying!
My immigrant Tarpey ancestor is:
Michael L. TARPEY b. Co. Galway ca. 1853, son of Patrick TARPEY and Mary HEALY. Arrived in New York about 1872. Married 18 July 1872 to Anne O'DEA, St. Teresa's Church, New York, NYC. Died Brooklyn, NY, 28 February 1917.