Yes! Carthage was indeed a republic, albeit an oligarchic one (as Aristotle suggests). Its government was similar to that of other Phoenician cities in the Western Mediterranean: two annually elected "judges" (špṭm in Punic, Latinized as suffetes) served concurrently as the city's chief magistrates, who presided alongside the Senate (’drm, the "Mighty [Ones]") on matters of legislation and diplomacy. According to Aristotle, the judges (whom he calls "kings) and the Senate could decide whether to submit proposals to the popular assembly (known officially as the ‘m qrtḥdšt, "People of Carthage"), at which point the assembly could debate, amend, and ratify these proposals as they saw fit; a particular idiosyncrasy was that if the judges and the Senate could not agree on whether to bring legislation to the People, decision-making powers automatically fell to the People. There is some evidence that the judges were popularly elected, perhaps through the assembly or some other process, whereas the Senate apparently replenished its ranks through co-option.

In practice, the politicians all seem to have come from the same social-economic class, and certain families were apparently able to monopolize political and religious offices generation after generation. One inscription (unfortunately undated) records, for example, that a chief priestess named Batbaal was the daughter of Himilkat the rb (an honorific title; he was probably a senator or general) and wife of Himilkat the špṭ, whose father, grandfather, and great-grandfather had also served as špṭm. Our sources also note that bribery and embezzlement were commonplace and that political offices could be bought. Thus, while the People wielded relatively broad authority in theory, this authority only applied if the judges and senators (who presumably shared the same interests) allowed it or if deadlock forced the issue down to the assembly.

I hope you find this helpful! :)