You can have 3 sacrifice flies in an inning if one of the sacrifice flies included an error that prevented the team from recording an out. This is similar to getting 4 strikeouts in an inning, where a missed third strike allows one of the batters to reach base on the strikeout.
There have been 4 times where a team had 3 sacrifice flies in one inning. Strangely two of those were by the Yankees in 2000, both with Andy Pettite pitching for the Yanks, and one by the Mets against the Yankees in 2004. The other time was by the White Sox in 1962.
In that August 19 game Angel left fielder Ron Gant dropped Jorge Posada's fly ball with a runner on third. Posada was credited with a sacrifice fly and an RBI, and he reached base on Gant's error.
Only if there was an error on one of the plays, the batter might get credited for a sac fly, but reached base on error.
In 1972, Wade Blasingame played in 12 games for the New York Yankees, batting in all of them. He had 2 at bats, getting 0 hits, for a .000 batting average, with 1 sacrifice hit, 0 sacrifice flies, and 0 runs batted in. He was walked 1 time. He struck out 2 times.
In 1915, Ed Barney played in 11 games for the New York Yankees, batting in all of them. He had 36 at bats, getting 7 hits, for a .194 batting average, with 1 sacrifice hit, 0 sacrifice flies, and 8 runs batted in. He was walked 3 times. He struck out 6 times. He hit only singles.
In 1930, Ownie Carroll played in 11 games for the New York Yankees, batting in all of them. He had 10 at bats, getting 2 hits, for a .200 batting average, with 1 sacrifice hit, 0 sacrifice flies, and 1 run batted in. He was walked 2 times. He struck out 1 time. He hit only singles.
In 1967, Steve Barber played in 17 games for the New York Yankees, batting in all of them. He had 29 at bats, getting 5 hits, for a .172 batting average, with 2 sacrifice hits, 0 sacrifice flies, and 4 runs batted in. He was walked 1 time. He struck out 17 times. He hit 1 double, 1 triple, and 0 home runs.
In 1936, Ben Chapman played in 36 games for the New York Yankees, batting in all of them. He had 139 at bats, getting 37 hits, for a .266 batting average, with 2 sacrifice hits, 0 sacrifice flies, and 21 runs batted in. He was walked 15 times. He struck out 20 times. He hit 14 doubles, 3 triples, and 1 home run.
In 1915, Walt Alexander played in 25 games for the New York Yankees, batting in all of them. He had 68 at bats, getting 17 hits, for a .250 batting average, with 4 sacrifice hits, 0 sacrifice flies, and 5 runs batted in. He was walked 13 times. He struck out 16 times. He hit 4 doubles, 0 triples, and 1 home run.
In 1974, Sandy Alomar played in 76 games for the New York Yankees, batting in all of them. He had 279 at bats, getting 75 hits, for a .269 batting average, with 4 sacrifice hits, 2 sacrifice flies, and 27 runs batted in. He was walked 14 times. He struck out 25 times. He hit 8 doubles, 0 triples, and 1 home run.
In 1932, Ivy Andrews played in 4 games for the New York Yankees, batting in all of them. He had 9 at bats, getting 2 hits, for a .222 batting average, with 1 sacrifice hit, 0 sacrifice flies, and 4 runs batted in. He was walked 2 times. He struck out 0 times. He hit 1 double, 0 triples, and 0 home runs.
In 1937, Ivy Andrews played in 11 games for the New York Yankees, batting in all of them. He had 15 at bats, getting 1 hit, for a .067 batting average, with 2 sacrifice hits, 0 sacrifice flies, and 0 runs batted in. He was walked 0 times. He struck out 4 times. He hit 1 double, 0 triples, and 0 home runs.
In 2007, Wilson Betemit played in 37 games for the New York Yankees, batting in all of them. He had 84 at bats, getting 19 hits, for a .226 batting average, with 2 sacrifice hits, 0 sacrifice flies, and 24 runs batted in. He was walked 6 times. He struck out 33 times. He hit 4 doubles, 0 triples, and 4 home runs.
In 1945, Hank Borowy played in 18 games for the New York Yankees, batting in all of them. He had 50 at bats, getting 11 hits, for a .220 batting average, with 3 sacrifice hits, 0 sacrifice flies, and 6 runs batted in. He was walked 3 times. He struck out 12 times. He hit 2 doubles, 1 triple, and 0 home runs.