Blink cameras are proprietary to Amazon. They only upload to the Amazon cloud. To be able to send images to Weather Underground you need a standard IP camera that lets you FTP images (not video) directly to Weather Underground's webcam servers. If you can manage to pull stills (not video) to your computer then you could use a program to FTP the images. There are a few programs that are able to run on your computer to do this. I'll let others chime in to make recommendations of which programs they are using to upload images. But your first order of business it how are you going to pull stills from the Blink system? It may or may not be possible. Sometimes individuals that can code come up with some crafty DIY solutions.
But that is going to be a complex system of parts; pulling the stills from the cloud, then having another program FTP them to WU. I think you'd probably fair better to get an standard IP camera that can do the job.