Calling a URL and parsing it for plain text values is not what you would want. What you first require is an HTTP API from the site owners which will provide you with proper values. But this certainly requires an API defined at their end and would require you to spare them some usage fees if at all the API exists.
Then a typical interaction between your program and their site would be something like this: You call their URL using either HTTP GET or POST (as mentioned by stephen84s) as specified in their API specification providing them the zip code and they return you the current whether for that zip code as a response.