Is there a way to backup a mysql database without having to run mysqldump?
Is there a way to run a backup of a mysql database using sql? 


Short answer: No. 


  ... scripts ... only have mysql user credentials, not system login credentials. 


I'm pretty sure it's possible to run the mysqldump utility against a remote database. 

However, I would have to ask "Why?" 

Even if you could take a backup, what would you hope to do with it? 

It seems to me that you don't have the tools or the credentials necessary to take proper backups (or to restore a database, which pretty much has to be done on the machine itself).  As such, I would guess that someone else is responsible for keeping these databases up and running, recoverable and well-tuned (at least I hope so, or your organisation is in Big Trouble).  

Perhaps you should discuss options with them?