Why do some communities daven late on Shavuot? Is that the preferable thing to do?
There is indeed a widespread custom to begin Shavuot late although it is far from universal and a relatively “recent” (in Jewish history terms) development. The Taz and the Magen Avraham both quote the custom of waiting until nightfall before starting Shavuot although there is a small difference between the two. The Taz emphasises waiting to begin Maariv whereas the Magen Avraham suggests that it is Kiddush that must begin after nightfall.
The custom is linked to the word “temimot” (תמימות) which the Torah uses to describe counting the Omer. The full phrase reads, “You shall count for yourselves … seven weeks, they shall be complete” (Vayikra 23:15), implying that there is an aspect of the counting of the Omer that must be full. Although we would normally bring in Shabbat or Yom Tov early (as outlined later), this custom suggests that the completeness is achieved by ensuring that the final day of Omer counting is not cut short through this early davening or Kiddush.
It is important to note that, while many of us are used to the custom of davening early on Friday night, it is not intuitive to daven Maariv and begin Shabbat while the sun is in the sky and we do so only because the Gemara (Brachot 27b) makes clear that it is acceptable. What is less clear is whether there are exceptions to this (e.g. Seder night when we try to begin after nightfall) and whether this applies to starting one’s Shabbat meal or just to davening and Kiddush.
Interestingly, there are other ways in which the idea of completeness is applied to the Omer. The Medrash uses the word “temimot” to explain why we count the Omer at night. Although one might have thought the mitzvah of counting would apply during the day (either because most Yom Tov linked mitzvot apply during the day or because the counting is specifically connected to the Omer offering which is brought during the day) the Medrash infers from the word “temimot” that we count in the evening to make the counted day (that follows) as complete as possible i.e. if we are counting day 6 then we want as much of day 6 as possible to occur after we have already counted so we count at night (the beginning of the Jewish day).
Additionally, the Behag famously uses “temimot” to say that if one misses a day of counting, one has missed out on the mitzvah of a “complete” counting and therefore does not count the rest of the days of the Omer. While the majority opinion is that one does continue to count the Omer, even after missing a day, our practice is to do so without reciting the bracha, in deference to this opinion of the Behag.
In the early twentieth century, Rav Dovid Tzvi Hoffman, one of the foremost halachic authorities of the time, wrote on this topic, in answer to a question received from the rabbi of Hamburg (partly noting how challenging the custom is in places further north). He comments that for the weeks to be truly complete one should make sure to begin the counting on the first night right at the start of the evening which is not a custom people specifically have. He adds on that if one is late in counting on the first night of the Omer this detracts from the holiness of the day, whereas starting Shavuot before the end of the Omer on day 49 would add to the holiness, rather than diminish it. He concludes that both options are legitimate (beginning at the normal summer Shabbat time or waiting until nightfall) and the key factor is that one’s intentions are how best to connect to the Almighty.
This concluding thought of Rabbi Hoffman seems to me to be the key. Each community should follow its rabbinic authority, who should act (as always) with best intentions in the way they believe the Almighty would hope for. Whether you will be starting Shavuot early or late, may it be a Yom Tov that deepens your connection to the Torah and builds your relationship with Hashem. Chag Sameach!
 The Pri Megadim and the Mishnah Berurah seem to follow the Taz which has led to this being the prevalent custom but many early and late authorities seem to side with the Magen Avraham. See, for instance, Masat Binyamin (Chiddushei Dinim, Orach Chayim 4), Shelah (Shavuot, Ner Mitzvah), Yechaveh Da’at 6:30, Tzitz Eliezer 13:59, among others.
 Torat Kohanim 12:6
 Responsa Melamed Leho’il, Vol. 1 OC 108
 He quotes this as the opinion of Rav Yaakov Emden, Mor U’Ketziah 489. Interestingly, he also notes that Rav Yaakov Emden in his siddur refers to the custom of waiting to daven until nightfall as “a weak, new observance from later authorities” (דקדוק קלוש מחודש מהאחרונים)!
 Paraphrasing the aphorism of our Sages, אחד המרבה ואחד הממעיט, ובלבד שיכוין לבו לשמים – One who does more and one who does less, as long as their heart is directed to Heaven.