Machen's "papers" - his correspondence, etc., are preserved at the Montgomery Library at Westminster Theological Seminary, in Philadelphia.
Grace Mullen is the librarian overseeing that collection, as well as the papers of other faculty members, and she would be the person to contact regarding the Machen papers. There are about fifty boxes in the Machen collection [25 cu. ft.?], though I don't know that a finding aid (index) has been prepared for that collection. As I understand, the collection is preserved [in good archival fashion] in the order that Machen had it in, namely by year. A "finding" aid would typically be a folder-by-folder list of what is in each box. An "inventory" would be an item by item list, with description. Inventories are rarely produced by archives anymore, except for the most important of collections.
There is a smaller body of Machen correspondence preserved at the PCA Historical Center, mostly between Machen and J. Oliver Buswell, Jr. An intern at the PCAHC prepared an "inventory" of that correspondence under my direction.
See
PCA Historical Center : Inventory of the Buswell-Machen Correspondence, 1926-1936 for details.
I think I did hear of plans to publish a book of selected letters, but I don't have further information at this point.