Question: I like to have folders in my theme folder for scripts and styles, but I can't access these folders in the "Edit Themes" interface. This is problematic for clients that don't give me FTP access. Is there a way to see these folders and the files in them from the Wordpress admin section?

Answer: 
For those looking for more recent information, **AceIDE** (forked from the now unmaintained WPide) works pretty well for quick and dirty edits.

Keep in mind that theme and plugin edits will be overwritten by updates, so editing this way is bad practice.
If a client won't or can't give you FTP access then you shouldn't be working for them. You could take the time it would take to hack the core and just find better clients.