Apple Notes handles the document side. Scan transcripts and certificates straight into a note with the built-in scanner, pin the note per school, check things off as they upload. Pair it with Files folders per university and you have a mobile app to track application documents without installing a thing.
If the built-ins feel too simple, the usual suspects all have solid iOS apps. Microsoft To Do for cleaner task views. Trello for the board people. Notion if you want a real database of programs, deadlines, and document status in your pocket. Google Calendar for the deadline alerts, with two reminders per date, because one is never enough.
The pattern matters more than the app. One list per school. Every document as its own checkable item, matched against the application requirements checklist for international students at your target programs. Weekly review on Sundays: open the list, open each portal, confirm received.
A graduate school application checklist fails in exactly one way. It goes stale. The app that wins is the one you open without thinking. Pick the one already on your home screen, an application checklist app for iOS or Android that you will not abandon by November.
Build your lists from real data, not memory. The requirements database at applicationreview.fyi breaks down document lists program by program. When every box is checked, get your application audited before you submit.