A handful of people have already endorsed Christine Love's work, all of which is emotional and I'm fond of dearly, but I'll chime in with Analogue: A Hate Story and, to a slightly lesser extent, Hate Plus, its sequel. The leading ladies have absolutely gut-wrenching stories that are worth seeing through to the end and the climax of Analogue is so powerful that I still cry every time I go through it despite playing through the game several times. Fantastic achievements in game writing all around with both of them.
Disaster Report and its sequel, Raw Danger, I feel are also really emotional powerful. Some people will argue that they're clunky games bogged down with dated mechanics and held back by the PS2 hardware, but they're so earnest about tackling the subject of surviving natural disasters without making a spectacle out of things that I can't help but love them dearly. Granted, I played them in Japanese and I mention that because the English edition was subject to some very strange localization changes that rob those games of that original identity, something that I can't imagine those games not having since they helped me cope with my memories of the Tohoku earthquake a few years back, but I know the games have their followings overseas still, so if you get the chance and can find them for cheap, it might be worth a shot.
You have a Chie icon, OP, so you might well have already played it, but I also found a lot of the side stories in Shin Megami Tensei IV to be touching. The main plotline is take it or leave it with a lot of people, but if you're one of those people that enjoys talking to NPCs and doing a lot of sidequests to get backstory stuff about the game, SMT4 will reward you in spades. You have to be pretty vigilant and talk to the same NPCs somewhat routinely since many of them ongoing plotlines that develop the further you get into the game, but there are definitely some in there that moved me quite a bit, especially when you get to the end of the game and everything is slowly coming to a head.
A lot of other great suggestions in this thread that I would echo, too, but even so, I'd still strongly suggest checking at least some of the games I've brought up here if you have time. :)
(Edit: Dammit, how could I forget Dangan Ronpa 2? The emotional payoffs at the end of the game will be bigger if you've already played the previous game, but having played it in Japanese, I'm not exaggerating when I say that Dangan Ronpa 2 is one of the best written and most emotional games I've played bar none. It's natural to go into the game expecting some pretty traumatic moments if you're already familiar with the original game, but 2 has such a profoundly lovable and relateable cast that when things go bad, you feel god damn terrible. There were a few scenes that were so hard for me that I actually had to put the game down and cry stuff out for a few minutes because of how intense it gets. I can only hope the English localization does it justice because the original Japanese is just superb and I love it so, so much.)