What government bodies were elected and by whom in Sakai Japan during the 500s and 600s AD?
Sakai did not exist as a geopolitical entity in your specified timeframe. Therefore, in one sense the "kind of political institution" that existed in 5th-6th century Sakai was of the non-existent variety.
The medieval city grew out of the earlier manors of Sakai, which debuted in recorded history as late as 1304. This was on a court document assigning ownership to Saionji Shōko. In fact, the name of Sakai (
堺) was first mentioned only in 1045 as
saka-i (さか井) in a poem of Fujiwara no Sadayori. It attained its modern form slightly later in 1081, as recorded in a diary entry by Fujiwara no Tamefusa:
申剋参住吉社奉幣 戌剋着和泉堺之小堂 住吉神主國元依罷神主清経送粮米等
... stayed the night at Sakai, Izumi ...
Clearly this vastly postdates the question's period of AD "500s and 600s".
Alternatively, let us approach the issue examining the area that later became Sakai. The literal answer to the question of "[w]hat government bodies were elected" is none that we know of.
Both the city and manor of Sakai straddled the border between Settsu and Izumi under the ritsuryō provincial system. In fact, the name Sakai itself is thought to derive from the word "border",
kuni-zakai (国境). During the question's timeframe (i.e. late Kofun and early Asuka), Izumi and Settsu were both part of the proto Kawachi province.
Our knowledge of the period is quite limited - the oldest extant history text in Japanese history dates only to AD 720 (though was begun in 681) and is heavily mythological. However, the 9th century Kujiki identifies the province as under the dominion of the Ōshikouchi clan
(凡河内氏), who apparently monopolised the position of Kuni no Miyatsuko
国造 and claimed descent from Hikokosohori no Mikoto
(彦己曾保理命). This makes them cousins to the Yamato royal house.
A Kuni no Miyatsuko were essentially a vassal of the Yamato king. They were appointed from regional lords who had pledged fealty to the Yamato court, and retained a high degree of internal autonomy. The Ōshikouchi Kuni no Miyatsuko specifically is thought to be in charge of continental immigrants and diplomacy. Little is known of their domestic political relationship with lesser regional elites, however.
The site of future Sakai appeared to be little more than a graveyard at the time. Since the Koffun period it has hosted the Mozu Kofun Gun, a series of tombs including the unreasonably oversized Daisenryo Kofun of (supposedly) the Nintoku Emperor.
To its south, there was a settlement named Suemura
(陶邑). As the name implies, Suemura was a major manufacturing centre of sue ware (i.e. Japanese pottery). It seems that the operations were centrally managed by the Yamato court, but there are little concrete details. Suemura is believed to have eventually became Izumigaokacho
(泉ヶ丘町) which was merged into Sakai in the 1950s.
There are no indication at all that they "elected" any "government bodies".
Ultimately, the question's assumption that "government bodies were elected ... during the 500s and 600s" in Sakai is totally specious. Rather than the 6th century, it seems the question really intended to ask about Sakai in the 16th century, circa the Azuchi-Momoyama period.