Adobe has slowly been rolling out third-party document-storage support for its free mobile PDF reader; for the chosen few on iOS, it meant the ability to directly open and save documents to services from big players like Microsoft and Dropbox. As of today, you'll be able to add those capabilities yourself with any service -- like we've been able to do on Android -- thanks to new support for iOS' document picker.
Previously, without direct support you could at best workaround the problem via the iOS share sheet (that rectangle-with-an-arrow icon). The document picker is Apple's other programming interface for allowing apps to talk to each other.
It does require a bit of setup; you have to manually add the relevant apps to Acrobat's list of services, dubbed "locations." Still, for heavy PDF users (like me), it may come in quite handy.