While you could do this with Tasker and SharpTools Tasker Plugins (for SmartThings), you’d probably be better off doing it as a Device Handler directly within the SmartThings platform. It looks like you’ve already posted on the SmartThings Community, so hopefully you get some traction on your question.
Is there a reason you specifically want to use the Weather Institute’s Open API rather than one of the already available weather devices for SmartThings?
For example, SmartThings has a built-in ‘SmartWeather Station’ which you can add using the instructions in the following post:
Weather Tile: How to Setup