There is no built-in way to download the Bricklink wanted lists. However, there is a userscript that should work for this.
DISCLAIMER: Use at your own risk. YMMV.
Firefox with Greasemonkey is the way I use other user scripts, but I wasn't able to get this working with Firefox. I was able to get it working with Chrome and Tampermonkey.
You'll want to make sure you read the description in the link above, and pay attention to the tips they recommend like increasing the number of items per page so you capture everything.