R David Brofsky (in his book Hilkhot Mo'adim, p. 573 and here online) brings sources on all sides of the debate. I have somewhat restructured the text to group opinions and added the sub-titles in bold.
(1) Those who argue one can pray before dark but need to wait for Kiddush
R. Yeshayahu Horowitz (1565–1630), the Shelah, writes in his Shenei Luchot Ha-Brit (Masechet Shavuot):
I received [a tradition] from my teacher, the Gaon R. Shlomo of
Lublin, who received [this tradition] person to person from the Gaon
R. Yaakov Pollack, [that one should] not make Kiddush and eat on the
first night of Shavuot until after the stars have appeared. The reason
is because it says regarding the counting [of the omer], “Seven
complete weeks there should be;” if one recites Kiddush while it is
still day, one slightly detracts from the 49 days of sefirat ha-omer,
and Shavuot is supposed to be [observed] after the [full] count.
This tradition dates back to R. Yaakov Pollack (1460–1561), the forefather of the Polish rabbinic tradition. Interesting, R. Horowitz writes that even though one may not recite Kiddush before nightfall, one may still recite the evening prayers early, as even on Shabbat, one may recite the prayers of Motzaei Shabbat on Shabbat.
The Magen Avraham (494), however, as well as the Peri Chadash (494), cites the Shelah, writing that one should not recite Kiddush until after dark.
(2) Those who argue one can pray and recite Kiddush before dark
R. Yosef Hahn (Frankfurt am Main, 1570-1637), a contemporary of R. Horowitz, records (Yosef Ometz 850) that he had not seen this practice in Germany. Furthermore, he argues this practice is not only an unnecessary stringency, but it also takes away from the time one could learn at night, as the night is relatively short during the summer. This seems to have continued to have been the practice in Germany thereafter as well, as R. Netanel Weil (1687–1769) writes in his comments to the Rosh, the Korban Netanel (Pesachim 10:2), that one may recite Kiddush and eat while it is still light on all Festival days, including Shavuot.
R. Yaakov Emden (1697– 776), in his Siddur Yaavetz, insists that, on the contrary, one should pray before dark in order to fulfill the mitzva of adding from the weekday onto Shabbat and Yom Tov (tosefet Shabbat).
(3) Those who argue one should wait until dark to pray
Although these early authorities only mention delaying Kiddush until evening, the Taz (494) records that the congregation delays beginning Arvit so that the count should be “complete.”
R. Shimon Sofer, in his Hitorerut Teshuva (56), suggests a different reason to delay Arvit; we should wait until night to ensure that even those who will stay up the entire night will not forget to recite Keri’at Shema after dark, its proper time. Similarly, R. Natan Gestetner (Responsa Lehorot Natan 7:31) suggests that Arvit is not recited until dark simply to ensure that people do not recite kiddush before dark. Numerous Acharonim, such as the Peri Megadim (Mishbetzot Zahav 494, s.v. me’acharin), the Shulchan Arukh Ha-Rav (494:2), the Kitzur Shulchan Arukh (120:11) and the Mishna Berura (494:1), rule that one should not recite Arvit until after dark.
R. Naftali Tzvi Yehuda Berlin, the Netziv, offers another suggestion in his commentary to the Torah (Ha-Amek Davar, Vayikra 23:21). The Torah says that one observes Shavuot “be-etzem ha-yom ha-zeh” – “this very same day” – in order to teach that there is no mitzva of tosefet Shabbat on Shavuot. We learn that we should observe Shavuot after dark from this verse, and not in order to ensure than our “count” is complete.