Is there a way to let users select a folder using a file picker package in Flutter and provide the app with read and write access to that folder?

No. User can select folder to get or write data from/to the folder but the app won't get read and write access. If user's required to select folder every time, you don't need any permission

If you need to read/write data from/to external storage without user action, you can use Scoped Storage but app's access will be limited to app specific external storage folder.