Replication of a remote MySQL database to local MySQL database
Check here (http://drcharlesbell.blogspot.cl/2011/06/comparing-databases-with-mysqldbcompare.html)

It explain how to Compare-Copy a DB to another DB, my case is the same as yours.

Depending on your access level you can:


Use a external client as middle man (to make de compare-copy)
Script in PHP to check new lines in the original DB and copy them
Use one of the many tools of MySQL to copy 


MySQLDBCompare
MySQLDiff
MySQLDBCopy