Allow uploads from widget control panel

I’ve been searching around, and I apologize if this is a duplicate. If it is, please point me in the right direction. I also apologize if this is under the incorrect forum.

Let me give you a little history so you might better understand my question.

I am the author of the Social Media Widget. Essentially, this is a widget that allows users to input URLs to their social media sites and select from different icon packs, animations, and icon sizes, then the widget will spit out the icons to the users’ sidebar.
https://www.ads-software.com/extend/plugins/social-media-widget/

A user approached me recently asking me if there were a way to input a custom icon (not just replace a current one, but in addition to the icons that are already available). I thought… why not?! Instead of making them upload the file from the media area, then inputting the URL to the image and the URL to the site and the image ALT tag, why not let them upload the image directly from the widget and then have the widget figure out the image part so they only have to deal with the URL to the site like with all of the other icons.

My question is simple (though I understand the answer is probably not):

How can I set up my widget to allow users to upload images via the widget control panel, and then how do I automatically generate the image URL and ALT tag from that?

Thanks for your help!

Brian