In 1872, Clapp began his professional career with the Middletown Mansfields of the National Association of Professional Base Ball Players. Over 19 games played, Clapp batted.278 with one home run and a team-high 30 runs scored while managing the team to a 5–19 record. After the team folded, Clapp joined the Philadelphia Athletics. His single home run tied him for the team-lead along with Wes Fisler, Cherokee Fisher, and Tim Murnane. Next season, in 1874, Clapp led the NA in at bats per home run ; his OPS was a career-high, while the Athletics finished the season 33–22, third in the NA, under manager Dick McBride. In his final year with the club, Clapp batted.264 with 77 hits and 39 RBI. His putout total was second in the NA among catchers. In 1876, Clapp joined the St. Louis Brown Stockings of the National League. He finished the year tied for the team lead in games played and hits, while he led the NL in putouts as a catcher, with 333. Next season, Clapp batted a career high.318, while his on-base percentage and on-base plus slugging percentages were the second highest in his career. In the field, Clapp committed 40 errors as a catcher, second highest in the NL to Lew Brown's 49. After leaving the team, Clapp joined the Indianapolis Blues, where he served as a player-manager for the 1878 season. Playing primarily in the outfield, Clapp was tied for the MLB lead in games played along with Indianapolis teammates Silver Flint, Russ McKelvy, Orator Shafer, and Ned Williamson. After his one-year stint with the Blues, Clapp joined the Buffalo Bisons. Playing in 70 games, Clapp managed the team to a 46–32 record, placing the Bisons third in the NL. On June 25 of that year, Clapp ended a streak of 212 consecutive games played, serving primarily as a catcher. In 1880, now playing and managing for the Cincinnati Stars, Clapp played in a total of 80 games, a career high, while leading the team to a 21–59 record. He played for the Cleveland Blues in 1881, when he earned the nickname "Honest": in May, a Chicago bookmaker named James S. Woodruff offered Clapp $5,000 to allow a passed ball with runners on base, and also wanted to be informed which games to bet on when the moves would take place. Clapp reported him to the Chicago police, which led to Woodruff's arrest. In 1882, after leading the NL in walks, Clapp made his minor-league debut for the New York Metropolitans of the League Alliance. In 1883, his last MLB season, Clapp played for and managed the New York Gothams for their inaugural season. Clapp, then 34, spent his final professional season with the St. Paul Apostles, where he batted.180 with 11 hits and a double.
After baseball
After retiring from baseball, Clapp served as a night sergeant in his hometown of Ithaca, New York. He died at midnight on December 18, 1904, of apoplexy. Clapp was interred at Lake View Cemetery in Ithaca.