Why is there no Mitzvas Hayom on Shavuos (For example, on Sukkos there is a commandment of shaking the Lulav, on Rosh Hashanh one must blow the Shofar, on Purim one must read the Meggilah, on Chanukah on must light the Menorah, etc.)?
Well, there were plenty of unique mitzvot in the time of the Beit Hamikdash, like bringing the shtei halekhem, bikurim, and of course the last of the omer. There are also sources that give the homiletic answer of Shavuot being the end of Pesach, like Shmini Atzeret is the end of Sukkot. S"A also has no stand-alone mitzvot. It is interesting that b'zman hazeh, we associate both with Torah: Matan Torah and Simchat Torah.
An answer from Chabad.org (I assume based on some talk or letter by the Lubavitcher Rebbe zt"l, although I haven't found the specific one) is that this demonstrates that the Torah is not just a set of laws superimposed on an existing world (i.e., that there exists a natural reality which Torah comes to regulate), but that on the contrary, the purpose of all of existence is for the sake of the Torah and its mitzvos. Thus,
Each year on Shavuot, when we re-experience Sinai, we show our appreciation for Torah through normal eating and celebrating—without any special rules. For the Torah does not introduce a new reality, but rather sheds light, purpose and sanctity into everything that is already here now. Even cheesecake.
Rivevos Ephraim 5:349 asks this question and answers in 6:284 that shavuos the Torah was given and all mitzvos are applicable on it(the mitzvah of shavous is keeping the Torah) other Yom tovim have specific mitzvos pertaining to itself.