There is a tradition to pay someone to say kaddish on behalf of people who have no sons to say it for them. R Chaim Binyamin Goldberg (Mourning in Halacha, p. 362) writes
If the deceased has no relatives to recite Kaddish for him, someone
should be hired to do so. It brings more merit to the deceased if the
person takes payment than if he does not not (Kaf Hachayim 55:30, Beit
Yosef end of 403, Ben Ish Chai Vayechi 15, Magen Avraham 132:2) In the morning, before praying, this person should state "All the kaddishim I recite today are for the benefit of the soul of [Ploni]"
R Eliezer Melamed in Peninei Halacha here rules similarly
When none of these relatives can say Kaddish for the deceased, part of
the inheritance money should be used to hire a God-fearing person to
recite Kaddish for him. It is good to hire someone who is engrossed in
Torah. If there is someone in the family who is occupied with Torah
study, he takes precedence over a stranger. The monetary compensation
for the Kaddish is important in order to ensure the fulfilment of its
recital. Furthermore, by employing someone who is involved in Torah or
a poor person who has children to support, the deceased will acquire
And indeed one can find a number of societies who will say Kaddish on behalf of departed Jews, e.g., Chevrah Lomdei Mishnah, Kupath Rabbi Meir Baal Haness and saykaddish.com
Anyone can hire someone to say Kaddish so I assume that who does it depends on the specifics of each situation.