The Rabbonim for the Karaimites operate the Beit Din service to oversee Karaimite needs. They consist of common Israeli Orthodox Krymchak and Subbotnik Jews in Judah ben Tabbai's Palestinian (Yerushalmi) Pharisee Minhag which is often called the Minhag of the Qara'im. 

One of the most ubiquitous misunderstandings in Karaite Judaism is that Karaims must reject the concept of Rabbonim and Oral Torah. That is rubbish. Rather they should say that we reject the Babylonian idea of an apostolic succession of Rabbinic authority through ordination from a club claiming descent from Sinai (or from Jethro in fact). 

According to Judah ben Tabbai's Minhag, all people whose Kalam is sufficient have the right to call themselves Rabbaonim. The idea of an Oral Torah descending from Sinai does exist in Judah ben Tabbai's minhag and is called Sevel HaYerushah which Isaac Lutski even said included the Lurianic Kabbalah. 

Karaimism includes Kabbalah and Sevel Ha-Yerushah and Jewish overseers as Rabbonim but lacks the concept of Rabbinic ordination by the authority of a rabbinic qualification from an institution rather than from Kalam. The only qualification for a Karaimite Rabban has always only been one of practical, applied, natural logic. That must always come first. Even knowledge of liturgy and scripture come later.