What a mess. First, as the situation is described, I think it's weird that Mom would get so upset about not getting a Christmas card from Sister. But, if that is representative of the level of seriousness with which the OP's family takes Christmas cards, then I think Sister was ultimately at fault for not seeing the task through herself. I'm picturing it as vital medication instead of a card, in which case it would make sense for Mom to be upset even after Sister explained the delay. Sister needs to change her delivery strategy, maybe paying for super premium overnight mail or something, or she and Mom need to change their agreement (stop expecting a Christmas card from Sister).
As for Kelly, well, I don't think she was being a good friend. I think her response is very telling, to come up with (far-fetched, IMO) excuses rather than saying, "I'm sorry, I can't/couldn't get them delivered on time." Maybe we would still be friends but I wouldn't consider her very reliable.
Story: A friend who lives out of town hinted that she would like a calendar from me this past December. I was happy to oblige and we made tentative plans to meet up when she would be in my town in early January. Then she decided that she would be too busy to meet with me, and the next week I was out of town for work. The next week (third week of January) I asked my dad to mail the calendar to her. He didn't get around to doing it until the next week (fourth week of January) so she didn't get the calendar until early February, by which point 1/12 of it was useless. I was aggravated at my dad, but at the same time, I realized it was my own fault for not taking care of it myself earlier (like as soon as I knew we wouldn't be meeting in person). It was more convenient for me to wait, and to have him do it, but I was taking a risk that he wouldn't do it on my schedule, and that indeed is what happened.