Were the Ten Commandments given on Shavuot?
Let's say for the sake of discussion that the people left Egypt on Nisan 15.
We are told, "In the third month after the children of Israel had gone out of the land of Egypt,
on the same day, they came to the Wilderness of Sinai." (Exodus 19:1)
So during Sivan they came to Mount Sinai on the same day of the week they left Egypt. This would have been Sivan 5.
The Jews observe Shavout on Sivan 6, but God told Moses to bring the people to the Mount on the third day.
"Then the Lord said to Moses, 'Go to the people and consecrate them today and tomorrow, and let them wash their clothes. And let them be ready for the third day. For on the third day the Lord will come down upon Mount Sinai in the sight of all the people.'" (Exodus 19:10-11)
Pentecost is fifty days from the Sabbath during Unleavened Bread, which would have been Sivan 8.
This is when God spoke his covenant to the people. The ordinances came later.