Hi,
Im not sure how weatherlink works, but either way, even if there was a possibility of data export as JSON or XML, since you want to use someone else´s site, I cannot think of how you could get that. Probably the only way would be parsing the HTML and by preg_match functions extract whatever data you need, which on the other hand isnt so difficult to do and if you only want to do it for personal use I dont see any major problem with it.