In the biblical account, the holiday of Shavuot seems to be a celebration of bringing bikkurim and matan Torah is not explicitly mentioned. Even in the Gemara, where matan Torah is said to have occurred on either the sixth or seventh of Sivan (Shabbat 86b) and the psukim about Shavuot are expounded such that Shavuot also falls out on the sixth of Sivan (Menachot 56a-b),

these two events are not explicitly connected. Why is this holiday of the first fruits nonetheless an appropriate time for us to celebrate our receiving of the Torah? Perhaps the beginning of an answer can be found in one of the first places that matan Torah and Shavuot are explicitly connected. The midrash first relates the drash from the Gemara about matan Torah being on the sixth of Sivan and then continues:

And on Shabbat which was the sixth of the month, they received the Ten Commandments and that was the holiday of Shavuot, the day of the giving of the Torah, the fiftieth day of going out of Egypt, that’s why it says ‘under a fruit tree I woke you up’ (Shir HaShirim 8:5) and ‘like a fruit in the forest’ (ibid. 2:3), as this fruit gives fruit fifty days after it blossoms, so the Torah was given 50 days after going out of Egypt. And the Torah was given at the send of even generations… 
Pesikta Zutra, Shemot 19:10

וביום השבת שהוא ששי בחדש, קבלו עשרת הדברות והוא יום חג השבועות, יום מתן תורה, יום חמשים ליציאת מצרים, לכך נאמר תחת התפוח עוררתיך (שה"ש ח ה), ואומר כתפוח בעצי היער (שם ב ג), מה תפוח זה מוציא פירות לחמשים יום משתפרח, כך התורה נתנה לחמשים יום לגאולת ישראל, ונתנה תורה לסוף שבעה דורות, אברהם, יצחק, יעקב, לוי, קהת, עמרם, ומשה, שהן סוף שבעה שבועות לגאולתן, שהוא יום שביעי המקודש, כי בו שבת מכל מלאכתו 
פסיקתא זוטרא שמות יט:י

Here matan Torah is dependent on going out of Egypt. It is not a one-day event, but rather a process contingent upon preparation. As fruits need cultivation and time to develop and blossom, so to our receiving the Torah is dependent upon study and cultivation. This idea that receiving the Torah is a process taking time, could be an explanation for the ambiguity of the date of matan Torah and its connection to Shavuot in the Torah. God could give us the Torah on one day, but our own receiving it and internalizing it is a process which takes much longer. The declaration given when bikkurim are brought emphasizes this idea of process: It doesn’t begin with picking the fruits or even planting the seeds, but rather with our ancestors going down to Egypt, becoming a nation, and then God bringing them out of Egypt and to the land of Israel (Devarim 26:1-11). It also emphasizes the fact that these fruits, which the bikkurimbringer has cultivated, cared for and harvested for months, ultimately come from God, “and thus I have brought the first fruits of the land which the Lord has given me” (Devarim 26:10). Like cultivating crops and bringing bikkurim, receiving the Torah takes work and investment but at the same time, recognition that our knowledge and understanding ultimately comes from Hashem. Perhaps we celebrate His giving it to us on one day, to remind ourselves to be engaged in the process of receiving it every day.