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?