I now may well be the only one who cares about any of this. I think I've now tracked down more genealogy on James Stodart than anyone else has put together.
In searching, I kept coming across another famous Stodart family. Out of Edinburgh, Robert and William Stodart were makers of fine pianos. These Stodarts had a shop about half a kilometer from James' shop on the Strand. The biographies I can find are really unsure what the relationship between Robert and William was. Nephew. Brother. Son. No one is quite certain. But Robert definitely made pianos, famously being the first to use the term 'Grand Piano' in a patent.
And according to Blackwood's Edinburgh Magazine, Volume 14 James Stodart died September 11th, 1823 in the house of his uncle, Robert Stodart. The same Robert. There is a plaque for James in the Old Calton Cemetary. His son David is listed on the plaque, but with no dates, so David is possibly not buried there.
Robert Stodart had a son named George who seems to have married James' daughter, Janet. From there the records become too tangled and contradictory. In any event, Faraday continued to correspond with George Stodart (who is listed as James' son-in-law in the 5 volume Correspondence of Michael Faraday.
In short, the Stodarts have six million individuals all with the same given names. This is difficult to sort out, but I've got some of it nailed down now.