Full WordPress Backup Plugin?

I have a scenario where I have admin access to a site, but no file system access. I need to get a full backup to move the site to a new host. Does anyone have any suggestions for plugins or procedures to get a full backup / copy of the WordPress site and files in this scenario?

Getting a copy of the database is easy, the challenging things is all the files. I would imagine there may be a decent plugin out there that does this, or I may be able to create a plugin to help grab the files.