I tend to read my ebooks on the Kindle app (I prefer the layout), and I usually send them to the app either through the iOS files app (if it's already a mobi file) or via the Send EPUB to Kindle website. The only issue is none of the books I load either of these ways seem to support continuous scrolling. From looking around the interwebs, it sounds like this is because that feature isn't supported for mobi files. Is there any way to sideload books in another format to the iOS Kindle app, or am I stuck without continuous scrolling unless I want to switch apps?
Continuous scrolling only works for books in KFX format.

I have had success sideloading KFX files to the Kindle for iOS app using iTunes. You cannot send them via email or using the Files app.