Is there a plugin that when upload from url/web, it will also give an option to store the images locally and/or into the media library?
I am afraid the upload through url does not uploads the image on your site server but it just inserts it in to the post so it can not appear in the media library.
Through a plugin
You can use the Grab and save plugin to achieve what you want but i think you don't need a plugin for this read the instruction below and you will understand why?
Without a plugin needed
However you can upload the image through a url and then insert it in the post so that it will also appear in your wordpress media library without the use of any plugin to do so follow the following instructions.
the image will be upload from the site and will now also appear in the media library
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