This is how I have my Gecko board hooked up.
Now for you here is how you would hook up the X-carve output to the VFD.
Because There is no octocoupler on the X-carve you will have to make sure that there will not be any reverse current that will stress out the X-Controller input. From the schematic it is tied directly to the pin. I would hook up a circuit that isolates the X-controller and the VFD. Here is a link to a single channel isolation circuit.
http://www.ebay.com/itm/80KHz-1Channel-Optocoupler-Isolation-Module-12V-No-Din-Rail-Holder-PLC-Processor/391758367386?_trkparms=aid%3D222007%26algo%3DSIM.MBE%26ao%3D1%26asc%3D20160908110712%26meid%3Dc2df6f5c6e324ca9b833a187fc32e568%26pid%3D100677%26rk%3D1%26rkt%3D17%26sd%3D352122708383&_trksid=p2385738.c100677.m4598
If you do use this optoisolation I would have to modify the schematic as you would need to to power it.
I hope this helps as I am not sure what you have and how your going to set up your system.
The PWM can be used to trigger if you so choose. Its just you have to figure out how to set up your G-code to output a constant voltage out that will represent a logic high. 2v or greater would be a logic high.
The MCU inside the X-controller can only handle 40ma of current flow from a any of the pins. Also you should be aware that the voltage reference for the PWM will be set in the firmware. You can measure the output by setting the necessary G-code setting for the PWM. use of the relays are not necessary. You could use a mosfett as well. Its up to you on what you want to do.