Depending on what you're looking for, a _lot_ more is being published by the Library of Congress as Linked Data nowadays, including LoC classification and subject headings. Check out https://id.loc.gov/.
Where do you go to actually find the Classifications or Subject Headings themselves, in whatever format?
That seems to have Subject Headings, but not the Classification, unless I'm blind (a fairly high likelihood).
I've spent quite some time poking around the LoC website over the years without finding anything more accessible than the PDFs of the Library of Congress Classification. Are you aware of that being generally available?