An emo card is short for 'emotional card', which can be two things.
A: A really sentimental card, either snail mail or ecard.
B: A metaphorical card that some pulls out to manipulate others in games of personal psychological warfare; a way of manipulating people using emotional tools. An emo card is an emotional tool.
A: Oh nice. Joe Butthead just sent me a big emo card telling me how fabulous I am. Too bad he addressed it to the wrong person!
B: Leticia Lonelyme saw she was not getting her way with Mr. Couldgiveadamn, so she brought out her emo card, and did a very convincing weeping Moment. Five minutes later, she was smiling inside as Mr C. gave in and did what she wanted. (hey, use your imagination folks, this is a family channel. Sort of.)