Hi,
You can use winHttp component that ship with winxp and above. It ships as a package called BITS [background intelligent transfer service]. Using this you can make calls to Http and Ftp servers. So this will enable your VB6 application to act as an HttpClient which will receive the data from the weather server.
Second thing is the server. You must know in which format you are getting the data from the server. Is it plain text ?, Soap ? or XML ? once you identify the format of data transport you can receive the data from the server using winHttp and parse them in your application and fetch the desired data.
Thanks.