Mom can be found inside her diner on the Den East Side. Talk to her about what's going on in town, and she'll ask the Chosen One to deliver a meal to Smitty. Agree and head out to the West Side.
The Chosen One will find Smitty in the junkyard on the far west end, past Lara's gang's house. Give him the meal and he'll trade the Chosen One a stimpak. If the One then heads back to Mom's, she'll offer them a meal that restores a few Hit Points. The Chosen One can decline it and come back later if they are not in need at the moment.
It's possible to fail this mission, if it takes too long (at least a day); Smitty will just go to Mom's. There's no penalty except missing out on a stimpak, some XP, and free food. It's possible to gain the XP anyway by talking to Smitty after this and selecting the correct dialog option.
Also, if one intends to finish this quest, do it immediately. The children are likely to steal the meal. Also, Mom will not speak to the Chosen One unless they take the meal to Smitty.