I see what you mean. 'to document' is simply to record a matter in written (or other) form. Since 'institutions are processing his appeals', that would indicate that the matter under review has well and truly already been documented!
Are we then to infer that the particular letter referred to was meant to document that he 'is having difficulty with the process'? Hardly seems likely.
Perhaps the writer should have phrased it something like: "Although you advise in your letter..."