During the 20th Century did the Democrats and Republicans actively work to prevent the emergence of a third party?

Well, first, we have to tackle the fact that in throughout the 20th century third party candidates played significant roles in presidential elections. Namely, the election of 1912 and the success of the Progressive party, or better known as the Bull Moose Party, which rose to prominence on the back of former president Teddy Roosevelt. Roosevelt actually finished ahead of Taft and the Republicans, gaining 88 electoral votes and 27% of the popular vote and finishing 2nd in the presidential race. Many historians believe this schism between progressive Republicans who broke off from the party to form the Progressive Party was a major reason leading to Wilson's victory (Wilson's victory with only 41% of the popular vote was the lowest percentage for the winning nominee in the 20th century). The Progressive Party would go on to win 5 seats in congress in the 1914 elections, and, would run respectable candidates for various gubernatorial races. Most prominent members of the Progressive Party would seek reunification with the Republicans after 1916, however, a new incarnation of the Progressive Party would form and ran Robert M. La Follette as a presidential candidate in 1924. While not as successful as Roosevelt's campaign for President in 1912, La Follette did win his home state of Wisconsin, and garnered a respectable 16% popular vote. Beyond this, there was also the popularity of the Dixiecrat party in the South during the 60s as well.

A good book to read about the emergence of third parties in the early 20th century would be Others: Third Parties from Teddy Roosevelt's Bull Moose Party to the Decline of Socialism in America. The decline of these third parties was not due to Democrat or Republican collusion, but rather, the forces of geopolitics, schisms within new third parties themselves, and power struggles that ultimately weakened many young parties that did not have the history of party over person.