The “w” refers to “west,” as in “Writers Guild of America, West.” There are two Writers Guilds, East and West. They are separate unions but work together on many issues.
Usually when writers refer to “the WGA,” they mean the West.
Presumably the w is lowercase because it is not part of the WGA’s name, but is used to inform the reader that the writer is talking about a specific West program or initiative.
You’ll often see the Guild use all uppercase as well: WGAW.