Contacts would be one of the easier things to transfer from a damaged phone since they are saved through a provider.
If you had your contacts saved via a third party provider such as Yahoo or Microsoft mail, you should be able to log back in with those login credentials as the contacts would have been saved on their side of the server.
If you had them saved via iCloud, you should be able to restore by logging back in via your Apple Account. If that doesn't work, you can export them from iCloud.com/contacts and send them to your device to save.