sound - Would a woodwind instrument still play in outer space?

They wouldn't work.

In a pipe, wind blowing over the fipple, or past a reed causes vortices which give lots of different frequencies. Then at the open end of the pipe (or open finger holes) the change to a fixed pressure causes most of the vibration to be reflected back down the pipe, setting up standing waves at various harmonics, which produce the tone. With no atmosphere in space, the air would rush explosively down the pipe, there would be no reflection, no establishment of standing waves, hence no tone formed.

If you put a microphone on the body of the instrument you would probably pick up a lot of noise, of rushing air. You would not get a tone. The air down the pipe would be moving quite fast. Don't put your head in the way!