It sounds like when you run wyUpdate as a regular user all the proxies and network credentials are already cached and set up to access the local or remote download site. However when you run wyUpdate in a LocalSystem user you're not using the correct proxy and/or HTTP credentials to connect to the website.
Firstly, if connecting to the site requires proxy access then make sure the proxies are either passed to wyUpdate or are configured for that user (LocalSystem).
Next, regarding HTTP credientials, you can just pass these in as part of the download site. For example:
http://username:password@yoursite.com/updates/%file%
You need to use wyUpdate 2.6.16.4 (or newer) for that to work.