To start off, I'd try a normal XMLHttpRequest()
If that doesn't work, this other stack overflow question has good answers about how to get started developing Windows gadgets. One of the links there should probably help.
How to get started with Windows 7 gadgets
[edit] more added
In your comment, by "SOP" do you mean Same Origin Policy? If so, then maybe gadgets enforce it - seems pretty lame if they do - I have no experience with gadgets, so I don't know.
Since you don't control the third party server, the usual things I'd mention (JSONP or CORS) don't immediately apply.
You could create your own web service which reaches out to the third party, but it'd be easier to set up a proxy server. Nginx makes this pretty easy.
Another option to consider is Yahoo! Pipes. You could create a pipe to read from the weather data and produce something else. Pipes has support for JSONP, which would allow you to circumvent the same origin restriction. In fact, it looks like someone has already done this.
Hope this helps!