Will a 9 speed Shimano Deore derailleur be compatible with my 13-32 tooth freewheel?
Your RD-M591 is a 9 speed rear derailleur, and has these ratings:
|High sprocket Max||12T|
|High sprocket Min||11T|
So nominally your high gears are not high enough, there's a reasonable chance it would work anyway. It could be a bit noisier and have slower shifting in the high/hard gears.
Your low gear of 32 is between 28 and 34, and will work fine at that end.
Do note this particular derailleur does not have a clutch. I'm not sure if there exists a 9 speed derailleur from Shimano with a clutch.