There has never been a Major League Baseball player whose name, or even nickname, was "Pilgrim." Wherever he is, John Wayne must be disappointed. Early in their history, before they adopted "Red Sox" for the 1908 season, the Boston team of the American League was referred to in print as the Pilgrims and the Puritans, but it was never official: Officially, they were the Boston Americans.
Nor has there ever been a player whose name or nickname was "Pumpkin." Nor has there ever been a native of Plymouth, Massachusetts to reach the major leagues.
But I found enough key names to fill out a starting lineup -- including a few tur-key names.
1B John "Stuffy" McInnis. While there's never been a player with the name, or nickname, of "Stuffing," the 1st baseman of the 1911 and 1913 World Champion Philadelphia Athletics was the best of those nicknamed "Stuffy." He batted .307 lifetime, collecting 2,405 hits. (He was a rookie, and not yet the starter, when the A's won the World Series in 1910.)
Honorable Mention to Ryan Howard of the Philadelphia Phillies, whose nickname is "Big Piece." Even though he's a good fielder, and the National League doesn't use the designated hitter, he can be my DH.
2B Octavio "Cookie" Rojas. I was looking for players named "Cook," or at least "Cooke," and found quite a few nicknamed "Cookie." Cookie Rojas was the best of them, twice a .300 hitter and 4 times an All-Star. However, he had lousy luck. As a young player, he was a member of the ill-fated 1964 Phillies. His last 2 years were with the Kansas City Royals in 1976 and '77. In '76, when the Royals clinched the American League Western Division title, he and his protégé, shortstop Freddie Patek, jumped into the Royals Stadium fountains together. But they lost the ALCS to the Yankees both times, and in '77, Patek, who made the last out, could be seen crying in the dugout, upset that Rojas would never play in a World Series.
SS Ewell "Turkey" Gross. Easily the weakest entry here, but he was the only shortstop I could find. Played 9 games with the 1925 Red Sox, batting just .094. "Turkey," indeed. "Gross," too.
3B Harold "Pie" Traynor. Apparently, he ate a lot of pie as a kid. But as a grownup, he batted .320 lifetime, had 7 100-RBI seasons, collected 2,416 hits, was the starting National League 3rd baseman in the first 2 All-Star Games in 1933 and '34, and was probably cheated out of some Gold Gloves due to the fact that they weren't awarded until 1957. He was a key member of the Pittsburgh Pirates team that won the World Series in 1925 and the Pennant in 1927, and he nearly managed them to a Pennant in 1938. He was frequently called the best 3rd baseman of all time until Brooks Robinson came along. He's in the Hall of Fame, and the Pirates retired his Number 20.
LF Norman "Turkey" Stearnes. Got the nickname because of the way he walked. A Negro League legend with the Detroit Stars, it's hard to say what he would have done against consistent major league caliber competition, but had he played, instead, for the Detroit Tigers, with Tiger Stadium being a good hitters' park, he doubtless would have had had some good power numbers. He's in the Hall of Fame.
CF Felix Pie. Pronounced "Pee-YAY," the name means "Cat Foot." Now with the Pittsburgh Pirates, he was previously a decent reserve outfielder with the Baltimore Orioles.
RF "Turkey Mike" Donlin. He got his nickname because of the expression, well-known in his youth, of a "turkey gobbler strut," as he seemed to strut when he walked. He played all 3 outfield positions (not uncommon in his era). He was kind of the Joe Pepitone of his era, a very good player (a .333 lifetime hitter, who led the NL in runs scored for the 1905 World Champion New York Giants, and had 106 RBIs for the Giant team that nearly won the Pennant in 1908), but he was more interested in being a star. He'd done some stage acting, and even married a very popular actress of the early 20th Century, Mabel Hite. Her death in 1912 took a lot out of him, and by 1914 he was through with baseball, as much by his own choice as that of those in charge. (He'd missed most of 1906, all of 1907, all of 1909, all of 1910 and most of 1911 on tour with his theater company -- it wasn't like he could go on TV or make movies at the time.) He was actually an original member of the team that became known as the New York Yankees, as he played with the 1901 Baltimore Orioles.
C Clarence "Yam" Yaryan. A backup at best, but he did hit .304 in limited appearances for the 1921 Chicago White Sox.
P Camilo Pascual. Nicknamed "Little Potato," he was a tough righthander from Cuba with maybe the best curveball of the late 1950s and early 1960s. In 1959 he won 17 games for the best Washington baseball team between 1945 and 2012. When the Senators moved to become the Minnesota Twins in 1961, the team got better, and so did his record. He won 20 in 1962 and 21 in 1963. An injury shortened his 1965 season, but he did get to pitch in the Twins' first World Series. He even won 25 games in 2 years for the reborn Senators (the team that became the Texas Rangers in 1972). Although his career record was only 174-170, 174 wins is nothing to sneeze at, and 3 times each he led the AL in strikeouts and complete games.
P Omar "Turk" Lown. A decent relief pitcher, he led the AL in saves in 1959, helping the White Sox win the Pennant.
P Richard "Turk" Farrell. Went 10-2 with 10 saves for a terrible Phillies team in 1957, and was converted into a good starter for the early Houston Astros (Colt .45's from 1962 to '65). In 1963, he won 14 games for a 2nd-year expansion team (losing 13). Won 106 games in his career (losing 111), despite playing only 1 season for a winning team, the 1961 Los Angeles Dodgers. Unfortunately, quite a few of the early Colt .45's died young, and Farrell died in a car crash in England in 1977. He was only 43.
P Steven "Turk" Wendell. Somehow, I think his nickname may have been short for "Jive Turkey." But he did have his moments with the Chicago Cubs, and the Mets would not have reached the postseason in 1999 and 2000 without him.
Honorable mention, collectively, to the Atlanta Braves and the Cleveland Indians. When the Braves, then in Boston, won the World Series in 1914, the Cleveland team, having lost their guiding force and their naming source with the retirement of Napoleon Lajoie -- they had been the Cleveland Blues and then, officially, the Cleveland Naps, seriously -- decided to ride the Braves' coattails, and became the Cleveland Indians. Yes, Native Americans did live on the shores of Lake Erie. And, yes, the man believed to be the first Native American to play in the major leagues, Louis Sockalexis, did play for a Cleveland team, the Spiders of the National League from 1897 to 1899. But both of those stories, which you may have heard as a source for the "Indians" name, are untrue: They named themselves after the Braves -- who were, themselves, named for their owner, James Gaffney, an official in New York's Tammany Hall political organization, a "Brave."