When upgrading an iPhone, the only way Report-IT reports will transfer to the new iPhone is if the new iPhone was restored from an iTunes (or iCloud) backup of the previous iPhone that had Report-IT on it. Report-IT should be on the new iPhone because it was restored from a backup, not a new App store download.
Prior to the backup being made, the “Backup Reports” option in the Settings or Configuration page of Report-IT on the old iPhone should be switched on. This is usually on by default, however it may have been changed and should be checked before a backup/restore operation.