Tony doesn't begrudge anyone else their enjoyment of the holiday season; he just wants to be left out of it. He's never really seen the point. Bucky's usually all for it, but a disaster at work that forced him to cancel his plans to travel with his family have left him feeling a little curmudgeonly. The two meet and bond over their shared lack of interest in the holidays.
When Tony unexpectedly finds himself taking care of a pair of kids who rather emphatically expect Christmas to be Done Right, he finds himself turning to Bucky for help.
Maybe Tony didn't have to leave the team but he sure felt like it. In the days after Ultron, he drifts, alone and a little lost, until Bucky decides that it's his job to look after Tony's happiness. And who better to help him than Tony's own AI?