I’m developing a sandboxed macOS app using Qt, which will be distributed via the Mac App Store. The app:
Monitors the clipboard to store copied items.
Overrides the paste function of the operating system via keyboard shortcuts.
Modifies clipboard content, replacing what the user pastes with stored data.
Have you tested this? Your first statement suggests this is something in progress, for future release. Your other statements suggest these features are already functional. #1 and #3 are doable, albeit tricky trying to do both at the same time. I would be very surprised if you could do #2 from an app. That's something the user would have to do.
Can a sandboxed app continuously read and modify clipboard content?
Sure
What entitlements are required?
None
What permissions should I request from the user to ensure that my app works?
That's going to be tricky. It sounds like App Review might not like this. In my experience, they seem to want an app to be fully functional without asking the user for anything extra. By that I mean they consider the full functionality of an app to be that which it can do without asking for anything extra. They may not let you claim any of that extra stuff in your app's description or screenshots.
Of course, I'm making assumptions and ***** generalizations here. I'm not App Review and I can't speak for them. All I can do is relate my own experience, as well as imagine what I would do if I were working for App Review.
I can tell you that App Review doesn't pre-authorize anything. You have to develop it all, submit, and hope they approve.
You also have to think maliciously. Imagine yourself as a developer doing evil things. Could your app facilitate that activity? In other words, look at your app objectively, from App Review's perspective. Is this an app that would bring value to our customers? (i.e. They are Apple's customers, not yours.)
And I think you should also think of the value the app would bring to you as the developer. I used to have a similar app in the Mac App Store. I pulled it years ago and nobody misses it. There are lots and lots of other clipboard apps. What's that Qt code base going to do for you in the future?
Compare all of that to a more straightforward app, that doesn't deeply involve itself in any system services, written in SwiftUI, for iOS and all other Apple platforms. Just something to consider.