According to baseball-reference.com, twenty two players have worn uniform #13 as a Yankee.
It is always interesting to look deeper into lists like this. A few notes:
Spud Chandler was the first Yankee to wear #13. He wore the number for only one season (1937, his rookie year). Chandler was better known for wearing #21 which he wore from 1938-1947. When Spud Chandler won the MVP Award in 1943, he was wearing #21
Lee Stine wore #13 in 1938, but he appeared in only four games early in the season for the Yankees. Those were the last four games of Stine's short (49 total games) big league career.
Cliff Mapes wore #13 in 1948, but again for just one year, and again as a rookie. He wore this number when he first came up, but also wore uniform #3 that year. Mapes was a Yankee from 1948 through 1951. In 1949, Mapes began wearing #7. Mapes was the last Yankee to wear #3 before it was retired for Babe Ruth. He was also the second to last player to wear #7 before Mickey Mantle. (Bob Cerv was the last player to wear #7 before Mantle.)
After Cliff Mapes, no Yankee wore #13 for twenty-two years before former Rookie-of-the-Year Curt Blefary wore the number in 1970 and 1971. Blefary was the first Yankee to wear the number for more than one season.
The first Yankee to wear #13 for a significant amount of time was Mike Pagliarulo who, after trying out uniform numbers 46 and 3, settled on #13 for the bulk of his Yankee career wearing the number from 1986 through 1989.
Jim Leyritz was the next Yankee to wear the number for a significant period of time. He wore #13 from 1993 to 1996 (after previously wearing #12 from 1990-1992). Leyritz was traded to the California Angels after the 1996 season. When he returned to the Yankees in 1999 and 2000, he also wore #13. Jim Leyritz was wearing #13 when he hit the big World Series changing home run in the 1996 Fall Classic:
Alex Rodriguez played 1,509 games as a Yankee. We wore #13 throughout his entire tenure in the Bronx. He appeared in seven All-Star Games in #13 and won two MVP Awards with that uniform.
No one has has worn #13 since A-Rod.
The following are all of the 22 players who wore #13 as a Yankee:
Alex Rodriguez (2004-2016)
Antonio Osuna (2003)
Michael Coleman (2001)
Jose Vizcaino (2000)
Jim Leyritz (1993-2000)
Jeff Manto (1999)
Mike Figga (1998-1999)
Willie Banks (1998)
Charlie Hayes (1997)
Gerald Williams (1992)
Torey Lovullo (1991)
Alvaro Espinoza (1990)
Mike Blowers (1989)
Mike Pagliarulo (1986-1989)
Keith Smith (1984-1985)
Bobby Brown (1980-1981)
Doyle Alexander (1976)
Walt Williams (1974-1975)
Curt Blefary (1970-1971)
Cliff Mapes (1948)
Lee Stine (1938)
Spud Chandler (1937)