I currently run Shimano SG-B-53/39 chain rings coupled to a 9 speed 13/23 cluster. If I was to change my cassette to a 9 speed 11/32 would I need to also change my derailleur?
As ojs says in comments - depends on the derailleur, but it's very likely that you would have to swap the derailleur to accommodate an 11-32 cassette. Older 9 speed groupsets were not designed for wider range cassettes, they had small gaps between ratios instead. You may be able to get away with a 12-28 though.
The derailleur specs you need to find are: max rear sprocket size and total capacity. Total capacity must be equal to or larger than (largest sprocket teeth - smallest sprocket teeth) + (large ring teeth - small ring teeth).