Which day of Shavuos are you asking about? On the first day, those who daven “netz” generally do not have what I would call a “leibidik” davening. “Second minyanim” would be what you are looking for. Many of my friends stay up to learn but go home and sleep from around 4:30 AM – 8:30 AM and then daven.
As noted above, you are unlikely to find many �leibidik� shachris minyanim on Shavuos…most of the potential “celebrants” are barely awake and feel fortunate they can daven with a modicum of kavanah.
Many Yeshivos have “yeshiva” over Shavuos. I believe most Yeshivos would have dancing at night after Maariv and before the night meals (that’s what we have when I was in Yeshiva).