Our non-profit has a repository that was custom built for us from a developer who has since moved on. We have various "types" of series to put posts under for different feeds, and suddenly new published posts, though accessible via their URL, do not show up on the feed(s) in the live site.
I have attempted to look at tutorials to look at the api documents search, and it finds 669 documents, but I don't know what else to look for.
You had the right reflex looking for the documents in your API browser, that's what you need to find! You just need the right query. You have a list here: Document API Technical Reference - Documentation - Prismic, so for example if the UID of the document you linked is film-finishing-fund, you'd have to go to your API browser and type: [at(my.[custom-type-of-film-finishing-fund].uid, "film-finishing-fund")]. Or if you have a type articles for example and the documents not appearing are all of this type, you can also do this query: [at(document.type, "article")]. If it shows up on the API browser, it means the issue is not with Prismic but with your website.
If the changes do not appear on your website, you can try out some of the steps on this post:
Let me know if that helps, and if it doesn't what you've tried and we can investigate further
