Maharik (Responsa, no. 30 (44 in some editions)) rules that you may say kaddish for a non-relative because the prayer is meaningful regardless of whom recites it. The question is can I accept compensation for this recitation?
I have not located an outright prohibition against paying someone to say Kaddish for you. However, according Rabbi Maurice Lamm, as cited, here actually refutes the Maharik's statement that this is meaningful!
Paying for Kaddish to be recited by a stranger is meaningless according to Maurice Lamm’s The Jewish Way in Death and Mourning (Jonathan David, 1969).
Kaddish is more than an incantation said to add bonus points to a soul. It is way for sons to show the holy impact of a parent’s life. To get rid of a Kaddish obligation by writing a check is a hollow effort. A stranger’s Kaddish robs the surviving children of the comfort brought by a mourner’s year of Kaddish recitation.
Lamm explains that is better to reach the goals of Kaddish through other means. Bring merit to a loved one’s soul by studying Torah. Read a portion of the Bible: the Chumash, a section from the Five Books of Moses, a chapter from the prophets. Study a Mishna, which speaks of the Torah’s laws and ethics (The word “Mishna” has the same letters as “neshama” - soul.) Commit to learning a daf, a page, of Gemarah each day.
Having cited this, though, I know that there are numerous organizations that suggest hiring someone to say Kaddish for you. One of them, saykaddish.com seems to be run by Chaba"d. See this article. The rabbi does state that it is not in the spirit of halacha, but he also says that there is no prohibition.
In merging the 2nd article with the first, perhaps one might argue that since the money goes towards tzedaka, which is meritorious for the memory of the dead soul, this might lessen the concern of doing this. However, in my opinion and based on my deduction from both articles, it is certainly preferable for relatives to say Kaddish themselves. Ideally, it should be sons, then daughters, then brothers / sisters or any close relative rather than a stranger. In short, make an effort to both learn and actually say the Kaddish rather than seeking the "easy way out" and hiring a stranger.
Rav Eli Mansour writes:
Rav Moshe Feinstein (Russia-New York, 1895-1986), in his Iggerot Moshe (Yoreh De’a 254), writes that the person must recite at least one Kaddish per person per day, and he must explicitly notify all those paying him that this is the arrangement. Hacham Ovadia Yosef, however, in Yabia Omer (vol. 8, Yoreh De’a, Siman 37), rules that we may apply to this case the concept of “Zikaron Ehad Oleh Le’kan U’le’kan,” such that one recitation of Kaddish can count for multiple people. Hence, one may receive payment from several individuals to recite Kaddish on behalf of their deceased family members, as his Kaddish recitation can indeed apply simultaneously to several different people. This ruling is codified by Rav Shemuel Pinhasi in his work Haim Va’hesed, where he notes that this was also the position of the Sedeh Hemed (Abelut, 154) and the Sitz Eliezer (7:49).