I dont have any experience with the 240g but blank adaptors do exist. The adaptor enables the blanks to cycle the action, but dont expect it to be perfect. As AAChang mentioned, it can get messy.
My experience being with the M249 fired with a BFA I would recommend a lot of lube before you fire it. I've honestly seen someone with a spray bottle of break free hosing down the inside of the weapon prior to firing.
Yes. There is definitely a BFA for the M240G, and it will cycle the action if blanks are used with the BFA. The Corps uses them. No clue where you'd buy one, though.