We used to do the opposite and bring our chairs and rent an umbrella with no problem BUT always sat in the back. I would imagine it would be the same with your own umbrella, you would not be allowed in the front.

That being said,we just returned from an extended stay and it was very windy but they never closed the umbrellas. In the past it was when a squall of rain came through that the Club O umbrellas would be closed.