You can download it directly from the author here: http://elm-chan.org/fsw/ff/00index_e.html
It will be compatible with energia, since energia just uses GCC to compile your sketches. However this method is not packaged as a library, so you will need to make changes to the lower level hardware driver to get it to connect with your SD card.
Also note this library takes up around 8kb of FLASH in it's minimal R/W configuration with directories. Enabling some extra features increases this size. You should be fine on the CC3200.
The link I posted has more detail, but you will need to implement your device control interface.
However if you've come from petietFS then you probably have most of these done.
disk_status - Get device status
disk_initialize - Initialize device
disk_read - Read sector(s)
disk_write - Write sector(s)
disk_ioctl - Control device dependent features
get_fattime - Get current time
The fatFS can just be added ontop of these and then you can make use of the API calls. fs_open/fs_read/fs_write etc.
I'm not sure if someone has packaged this up into a energia library yet. but it doesn't take long to get it working.