every act of civil disobedience can be construed as a more petty crime, if you squint. i believe gandi was guilty of loitering but that's not the salient part of his action, philosophically speaking.
Personally, I have no idea what motivates Anonymous or if they feel morally obligated to act. It's just as likely that they believe they can commit a crime without penalty. The actions of the FBI are, in part, to deter future individuals from participating in Anonymous coordinated attacks by reminding people that they are not that anonymous and there are consequences to one's actions.
Gandhi, King and other activists all were willing to suffer the consequences of their actions. They were motivated by a principle higher than blind obedience after all.