The snapshot feature won’t help you out here.
You will want to get a full backup from them. Then each day have them self you a differential backup. Restore that differential backup over the full. Each time they take a new full you will want to have them send that to you.
After you do each restore make sure to leave the database is a readable state but where you can still apply a new backup to it.
You can also do this with transaction log backups as well. If you use the transaction log backups there are more files to deal with each day, but the nice thing is that you only have to get the full backup once.