If you're looking to start, I recommend giving a regular XMLHttpRequest() a try first.
If that doesn't do the trick, check out this other stack overflow thread with helpful tips on getting started with Windows gadgets. One of the provided links should point you in the right direction.
How to get started with Windows 7 gadgets
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In reference to your comment about "SOP," are you referring to Same Origin Policy? If so, it's possible that gadgets enforce this restriction - which could be frustrating. Without experience in gadgets myself, I can't say for certain.
Given that you don't have control over the third-party server, typical solutions like (JSONP or CORS) may not immediately apply.
You could set up your own web service to interact with the third party, but using a proxy server would likely be simpler. Nginx offers an easy solution for this - here's a guide: Nginx Proxy Setup.
Alternatively, consider utilizing Yahoo! Pipes. You can create a pipe to retrieve weather data and transform it as needed. Yahoo! Pipes supports JSONP, allowing you to bypass same origin restrictions. It appears someone has already created a relevant pipe you can explore: See Here.
I hope this information proves useful!