If that particular website provides a ‘web widget’ or otherwise allows embedding their site into another site/page, then perhaps you could use a URL Custom Tile to embed it. (I also put together a simple Refreshable URL Custom Tile for sites where the data doesn’t automatically refresh)
Otherwise there’s a few other Weather Options include a Weather Custom Tile – it doesn’t have the neat graphs, but it does have a variety of nice layouts you might find helpful.
Weather Custom Tile