You could take a look at dumping to a file on the server using the mysqldump utility, this can be done either by SSH or from a PHP script.
For example in SSH:
You could also do this through a PHP script using the system() call.
mysqldump -h localhost -u USERNAME -pPASSWORD --single-transaction DATABASE > /home/USERNAME/public_html/mysql_dump.sql
The only reason I'm suggesting this is because I'm not sure if phpMyAdmin can dump straight to disk, or if there are other web-based tools which can help you with this.
Just my £0.02p anyways