A little history that bears on the origin of Wanderin':

The "Near East" reference is probably to the efforts of what was later known as the Near East Foundation, which is still in existence today as an NGO.

From Wikipedia article and the Near East Foundation website: Founded 1915 for Armenian and Syrian relief, it was a response to the Armenian Genocide. First effort of its kind, it had nearly 1,000 U.S. citizens overseas building orphanages, vocational schools and food distribution centers. It saved the lives of nearly a million Armenian, Assyrian, Greek, Arab, Persian and other refugees from 1915-1930.

Unfortunately that doesn't help us date the song. I hoped it would. The only indications: the "Near East Relief" name was adopted in 1919.