Oh man, if you like stalking the download button as much as I do, here's the practical scoop: most works on Archive of Our Own let you grab an EPUB right from the work page, but whether a direct download shows up is up to the creator. On a typical fic page you’ll see a small 'Download' link or button near the top or in the chapter navigation; that will usually produce an EPUB for the entire work (multi-chapter or single). If the author has explicitly disabled downloads, that link won’t appear — and I always respect that choice, because some writers prefer control over distribution.
When I want a Kindle-friendly file I almost always grab EPUB and then handle conversion locally. My go-to is Calibre (free, reliable) or the FanFicFare plugin that plugs into Calibre — it can fetch AO3 works directly and output EPUB, MOBI, or AZW3. Another neat trick: if you prefer a clean single-file collection (say a series of related works), you can combine EPUBs in Calibre or use small scripts that stitch chapters into one book. Some authors also upload zipped MOBI/EPUB themselves in the work’s notes or a pinned link, so always glance through author’s notes and the endnotes for alternate downloads.
A couple of UX tips: use the page’s find (Ctrl/Cmd+F) for 'Download' if you don’t immediately see it; check each chapter page too — sometimes chapter pages have their own download links. And if you’re planning to read on a Kindle, converting EPUB to MOBI/AZW3 with Calibre usually gives the best layout. Most importantly, respect the creator’s settings and, if in doubt, politely ask — many authors will gladly provide a personal copy if asked nicely.