Why did master Oogway vanish along with a whirl wind of peach tree leaves in Kung Fu Panda (2008)?
And, why did he vanish just after the news of Tai Long's escape?
In movies, the deaths of spiritual masters are sometimes shown as a "transcendence". One example is Obi-Wan Kenobi, in Star Wars: A New Hope. As he was cut down by Darth Vadar, Obi-Wan's robes simply fell to the floor, empty. His body had vanished.
Master Oogway passed away in a similar fashion: He was surrounded by a swirl of peach tree leaves, representing the chi of the universe. He became one with it, and his body vanished (or became chi) as he died / transcended.
I think the answer to the 2nd question can be found here.
If you go through the link, it says that Master Oogway developed the legend of the dragon warrior. When Po was finally chosen as the dragon warrior, his quest or rather his purpose was complete.
I think the news of Tai Lung's escape and his vanishing was a mere coincidence.
PS: Also the link says that his (Master Oogway's) time had come to an end hence he ascended to heaven. But I think it was more to do with him completing his quest than just that (his time coming to an end).