I tried tried the link on the computer I'm using but couldn't get it to work (this system is sort-of "Internet armored" so I'm not surprised). However I could divine that the Star Wars Soundboard uses .swf (ShockWave Flash) files -- typical for this sort of thing.

After playing all the sounds (or after playing each one), the .swf file(s) of the audio should be in your browser cache. Grab those files and extract the audio using anyone of a number of freeware programs that do that sort of thing. I've used "Fortop SWF Resources Extractor" on occasion, with success. If this new version doesn't allow "operational testing", there is an older freeware release available -- like on the Only Freewares website:
http://www.onlyfreewares.com/Web-Development/Graphics-Editors/Fortop-SWF-Resources-Extractor.php

Worse coming to worst, play the file and record it with your system's sound recorder (save as a .wav file for no additional compression artifacts).