Are there any private schools in or around Wingdale?

There aren't any private schools in Wingdale itself. Your nearest and best options off the top of my head would be:

-South Kent School in Kent, CT
-Millbrook School in the Town of Washington
-Dutchess Day School in Millbrook (Town of Washington)

Some of the aforementioned schools are exclusive. If you're looking for Catholic schools, there are several elementary schools and a high school in the Poughkeepsie area, with the nearest elementary being St. Columbia in Hopewell Junction. The closest Catholic high school may be Our Lady of Lourdes in Poughkeepsie.