The core of the sources addresses the pasuk directly: the prohibition of afflicting the orphan and widow in Shemot Rabbah 30:8 and detailed in Sefer Mitzvot Gadol, Negative Commandments 8 and Sefer HaChinukh 65:5, which read 'כל אלמנה ויתום לא תענון' as establishing a special covenant (ברית כרותה) between God and the orphan and widow, such that their cry is answered directly, as the pasuk states שמוע אשמע צעקתו.
Both Sefer Mitzvot Gadol, Negative Commandments 8 and Sefer HaChinukh 65:5 qualify the prohibition: affliction for the sake of the orphan's own benefit—such as a teacher disciplining a student in Torah or a trade—is permitted, though even then one is commanded to treat them with greater mercy than others, and the status of 'orphan' for this mitzvah applies only until the child can manage his own affairs like an adult.
Shemot Rabbah 30:8 develops the theological stakes of the prohibition, explaining that God is called 'the father of orphans and judge of widows' (אבי יתומים ודיין אלמנות) precisely because their eyes depend on Him alone, so that one who robs them is considered to rob God Himself and draws down the wrath described in the continuation of the pasuk, וחרה אפי והרגתי אתכם.
Shulchan Arukh, Yoreh De'ah 251:1 addresses a related but distinct area—the laws of whom to give charity to and in what order of priority—rather than the prohibition against afflicting the orphan or widow itself.
The Ramban (Shemot 21:15) discusses a different pasuk entirely, concerning the death penalties for striking or cursing one's parents, and does not bear on the orphan-widow prohibition.
Likkutei Sichos 29, Shoftim likewise treats a different subject, the inner meaning of the mitzvah of mishpat (justice) and its connection to divine pleasure, rather than the specific prohibition of Shemot 22:22.
Chullin 94a:5-7 discusses the prohibition against deceiving people, including gentiles, a separate halachic category from the affliction of orphans and widows.
Among the responsa addressing practical application, Mishneh Halachos (Mishneh Halachos 12:156) rules that one who can support himself should avoid relying on charity since accepting it is a disgrace for a self-sufficient person, while someone genuinely unable to support his family must accept it rather than place them at risk, a position he supports by citing teachings on avoiding charity and gifts and the example of a righteous woman who refused wartime relief.