The Chicago Cubs play in the National League, and play at Wrigley Field. Wrigley Field opened in 1914 with the Federal League Chicago Whales, and the NL moved in when the FL folded after the 1915 season.
Officially, the Chicago National League Ball Club (Cubs has only been part of their official name for less than 10 years) was founded in 1876. But let's look at what happened between 1915 and 1916.
When the Federal League folded, former Whales owner Weeghman bought the NL Cubs.
The team played in Weeghman field, as had the Whales. The park was later named for the successive owner Wrigley.
The Manager was Joe Tinker, who had managed the Whales for two seasons. The Cubs acquired Tinker along with 11 players on February 10th.
Looking at the 8 regulars for 1916 described by Retrosheet, 4 were with Chicago NL in 1915, 2 were with Chicago (FL), one was acquired in a trade mid-season, and one made his major league debut in 1916. That edge would go to the 1916 team being the continuation of the Cubs.
On the entire pitching staff (11 pitchers), 4 pitched for the Whales in '15, 3 were with the Cubs (including George Pierce, who appeared in only 4 games in 1916), and 4 were elsewhere in the majors. This would tend to suggest that the 1916 team was the continuation of the Whales.
Looking at all position players with 100 plate appearances, we find 7 on the 1915 Cubs, 5 on the 1915 Whales (including Art Wilson, who began 1916 in Pittsburgh), and 4 who were either out of the majors or with another club in 1915. So this puts it at 9 Cubs and 8 Whales counting regulars and pitchers except for Pierce and Wilson, with 8 spending their 1915 elsewhere. I'd say this might be too close to call.
If you look at the 18 players who had less than 100 PA and who were not pitchers, however, we find only one player who was on the 1915 Cubs -- catcher Bob O'Farrell, who played 2 games in 1915 and 1 in 1916. There are 5 Whales, including Tinker, and 12 who were elsewhere in the majors or minors in 1915. Counting all players, this makes 11 Cubs and 14 Whales including the manager.
So, is the current team the continuation of the Whales or of the Cubs?
Looking at 1998, we find the Milwaukee Brewers changed leagues but kept their same ownership and players. They also kept the history of the 1970-1997 Brewers. In 1972, the Washington Senators of the AL East moved to Texas and the AL West as the Rangers. They retained legal rights to the name Washington Senators and the records of all the Senators; eventually they transferred the 1901-1960 history to the Minnesota Twins, who had previously been the Senators.
If the current Cubs are in fact heirs to the Whales, then they have never won the World Series. Granted, the pennant they won in 1915 did not lead to a berth in the World Series, but they still did not win the Series. They were only 20 games over .500 (86-66), and had 2 fewer decisions than the St Louis Terriers (87-67). The Boston Red Sox finished 51 games over .500 and the Philadelphia Phillies were 28 games over .500 that year