In the English "the outcry of Sodom and Gomorrah" it is as though Sodom and Gomorrah are the agent of the crying event. In the Hebrew this is less clear. The phrase is a simple noun chain with "outcry" in construct state. This can indeed indicate a genitive relation, but it can also be understood as "the outcry about Sodom and Gomorrah" → "the outcry against Sodom and Gomorrah". This is also how the NIV has translated 18:20 and 19:13:
18:20 Then the LORD said, “The outcry against Sodom and Gomorrah is so great ...
19:13 ... The outcry to the LORD against its people is so great that he has sent us to destroy it.”
This reading is supported by 18:21, where God unambiguously says the outcry has reached him:
אֵֽרֲדָה־נָּא וְאֶרְאֶה הַכְּצַעֲקָתָהּ הַבָּאָה אֵלַי עָשׂוּ כָּלָה וְאִם־לֹא אֵדָֽעָה
I will go down now and I will see if according to her outcry that has come to me they act entirely — and if not, I will know.
A 2002-3 article by Timothy D. Lytton in the Journal of Law and Religion, "Shall Not the Judge of the Earth Deal Justly?": Accountability, Compassion, and Judicial Authority in the Biblical Story of Sodom and Gomorrah, further explains this (pp. 36–37, references in footnotes omitted):
This ad hoc trial is occasioned by an "outrage" and an "outcry" emanating from Sodom and Gomorrah in verses 20 and 21 respectively. These two words in Hebrew, za'akah and tza'akah, are really identical terms, dialectical variants of each other (ref.), denoting a cry of distress or suffering. They also serve as technical legal terms for a claim of injustice linked to a demand for redress (ref.). The phonetic similarity between the Hebrew terms tza'akah (outcry) and tzedakah (justice) symbolizes linguistically that cries of disress are usually opportunities for doing justice (ref.). This wordplay works in two distinct ways. First, tza'akah is linguistically perverted tzedakah, reflecting that it comes from perverted justice. Second, tza'akah cries out, so to speak, for tzedakah (ref.).
Thus, "the outcry of Sodom and Gomorrah" is the distress over the injustice done there and the cry for justice to be done. Who is doing the crying is not mentioned. I would say it is the earth and injustice itself which cries for justice to be done.
On the other hand, as you mentioned, in Gen 4:10 it is the victim who cries for justice. But as the above shows, that is not crucial to the root. Crucial is the malformed justice on the one hand and the need for justice on the other.