I intend to show user a document selector just like the photogallery where user could select their icloud files and can download or make it available within the app. I would then use the files to upload on my custom API server.
Is that possible and how?
What you need is the native Document Picker. Take a look to the documentation
Basically you need to enable iCloud and iCloud documents on your application and then display the document picker. In there the user can select any file he wants. And by implementing the delegates you can retrieve the file and do what you want with it.