WPA stands for the Works Progress Administration. During the Great Depression, President Roosevelt instituted the WPA work-relief program to aid displaced workers.
According to estimates, more than 8 million people benefited from the program. Although the wages were low, the program allowed the unemployed to participate in socially-beneficent projects. Many worked on building new bridges, parks, airports, and roads. The WPA also supported thousands of artists who worked on creating murals and sculptures for federal and state buildings across the country.
Although Mr. Cunningham could have had a WPA job, he preferred to concentrate on farming. In the story, we know that part of Mr. Cunningham's land is entailed. This just means that part of his land has been set aside for his heirs and therefore, cannot be tilled or used for farming. The text tells us that "the acres not entailed were mortgaged to the hilt." This means that Mr. Cunningham owes the bank money for this portion of his land, and he also has to pay interest on his debt.
If Mr. Cunningham takes a WPA job, his land will be neglected, and he will lose the portion of his income that comes from the land. Also, while the WPA job is temporary in nature (the WPA was disbanded in the summer of 1943), Mr. Cunningham's land remains a permanent source of income for him. Despite the fact that he makes little money from his land, it affords him an independence that he cherishes. So, this is the main reason Mr. Cunningham will not work for the WPA.