If the users are in a domain and the backup needs to be agentless, you could run a logoff script when they log off that will copy the files to a share.
xcopy /e /y /i "%homepath%\Favorites" "\\server\share\%computername%_backups\favorites"
xcopy /e /y /i "%appdata%\microsoft\signatures" "\\server\share\%computername%_backups\signatures"
xcopy /e /y /i "%appdata%\microsoft\templates" "\\server\share\%computername%_backups\templates"
xcopy /e /y /i "%appdata%\mozilla\firefox" "\\server\share\%computername%_backups\mozilla\firefox"
This is a quick and dirty batch file that you can run when a user logs off that will copy their IE favourites, signatures, templates and Firefox bookmarks to a share. This script will create a folder called COMPUTERNAME_backups and embed the folders within it.
Just note that with this method the user will need to have write access to the folder, so I would recommend backing it up to a file server that is backed up regularly in case someone should find the share and tamper with it.