"Backing up" server tables to a local file is fairly easy. You can just use an append query into a local Access database for each table.
However, this is not a backup to recover from! A real database backup must adhere to consistency as any other database transaction. With the above approach you would only get a consistent backup if you would backup all tables in a single transaction in repeatable read mode. This is probably not possible with linked ODBC tables in Access.
I would instead suggest, you rather use
MySQL backup tools, which may also work on the local computer. The linked text also has lots of information on how to recover from the created backups.