What obstacles/challenges did FDR experience in getting the work progress administration started or keeping it going?

The major challenge Franklin D. Roosevelt experienced in keeping the WPA going was the 1936 Strike.

Explanation:

The strike which lasted for about 44 days in 1936 and 1937, forced the management of General Motors to come to terms with the striking workers, some of whom had just been employed by FDR in his Work Project Administration project.