the idea that Pentecost is Shavuot is from traditional Christianity, but is not Biblical.
1. According to traditional Christianity ascension day is 40 days after Jesus rose from the dead. This is based on a verse in Acts saying Jesus appeared to the disciples 40 times, and the 40th time he ascended. This proves that ascension day is not 40 days after after resurrection day because we are told in John that the 2nd time Jesus appeared to them was 7 days after the first time. And the third time Jesus appeared was when Peter said "I go fishing". Therefore at the very least Ascension day is 47 and more likely 50 days after the resurrection which would make ascension day Shavuot.
2. All the Jews were required to be in Jerusalem for Shavuot. If the Jews were already in Jerusalem at the time of Jesus ascension and it was just a couple of days before Shavuot there is no need for Jesus to tell them to remain in Jerusalem. On the other hand if Jesus ascended on Shavuot and Pentecost was fifty days later then it would make sense that He told them to remain in Jerusalem.
3. On Pentecost people joked that the disciples were drunk on new wine. That is impossible on Shavuot. You are fifty days away from having new wine. All the new wine from the previous year is gone and would no longer be called "new wine". However, 50 days after Shavuot is the feast of new wine. This joke would make perfect sense if this took place on the feast of new wine, rather than Shavuot.
4. Pentecost is the birth of the church. Jesus said you put new wine into new wine skins. The Spirit is the new wine and the church is the new wine skin. That word by Jesus points to the birth of the church taking place on the feast of new wine.
5. Peter tells the people joking about new wine that they can't be drunk on new wine because it is 9am. That comment would make no sense on Shavuot. However, on the feast of new wine the priest pours out the new wine onto the earth at 9am and blesses it, after that you can drink it. This comment only makes sense on one day of the year, the feast of new wine.
6. The feast of new wine is on the 8th of Av. The 9th of Av is when the temple is destroyed in 70 AD. About six months prior to Jesus crucifixion the Jewish leaders proclaim that Jesus is casting out demons by Beelzebub the prince of the demons, that was an official ruling. In response to this Jesus tells them that this ruling is an unforgivable sin. If he said this on the 9th of Av it would be exactly 40 years to the day of this ruling by Jesus that the temple is destroyed and cast down brick by brick. That would be according to the sign of Jonah saying in 40 days (years) this city will be cast down.
7. On Pentecost the disciples begin to preach the gospel telling the people of Jerusalem to repent for killing Jesus. On the 9th of Av in the Torah we learn that 3,000 died, at Pentecost 3,000 are saved. This becomes the basis for Hebrews to compare the two covenants, one given on Mt. Sinai and one given on Mt. Zion.
8. Every aspect of Jesus life: His conception, His birth, His death, His resurrection all took place on appointed feast days. He was conceived on Hanukkah, born on the Feast of Trumpets, circumcised on the day of Atonement, crucified on Passover and rose on the feast of firstfruits. Do you really think He ascended on a non feast day? Shavuot describes Moses, a type of Christ, ascending the mountain to God, a shadow of the ascension of Jesus Christ. If Jesus ascended on Shavuot and the Spirit was poured out on the feast of New wine that would fit much better with God who tells you the end from the beginning.
9. In Ephesians Paul tells us that the mystery hidden from the ages had been revealed. In Revelation 4 we see Jesus Christ as a lamb having just been slain, coming in and taking the scroll and opening it in the heavens. This is what Shavuot is all about, the scroll being opened up and given to God's people.