If I put a ping pong ball in a vacuum, would it pop? If so, at what point would it happen? Any standard table ping pong ball is acceptable.
Probably not - but without knowing the details of the specific ping-pong ball it's hard to tell.
The interesting part about the question is scaling laws. In a vacuum there is a pressure of one atmosphere (15psi or 10tons/m^2) acting outward on the inside of the ball.
BUT there aren't all that many square meters (or even square inches) on a ping-pong ball so the actual force is really quite low.
A sphere is a good shape for resisting this force, the most likely failure is the glue seam that holds the two halves together. And a ball has to survive the force of being hit pretty hard by a bat so the glue line is reasonably strong.
The answer to your question is "no". A quick web search for "ping-pong vacuum" yields a wealth of experimental evidence, in the form of articles with photographs about cannons that operate by putting ping-pong balls in vacuum and selectively releasing the vacuum.
Here are some videos of demonstrations. Note that the balls don't pop before being fired.