Drag and Drop PDF to Web Browser from Adobe Acrobat Window

I am developing a web application that captures multiple certificate based signatures (signed using a CAC card) on a pdf form and stores the document into a database. Microsoft Edge browser is the only browser available on the client machines. Adobe Acrobat Pro is also available on the machines. Currently, users have to download the PDF from the browser, open it in Adobe Acrobat, sign it, save it, find it in their saved location and drag and drop it into a browser upload area on the web page. Ideally, they would sign it in the browser and hit a button to save it back to the database (or maybe even not hit a button - just sign and let the application save it back to the database). I don't think this is currently possible using Adobe's PDF extension in the web browser. I've Googled this more than I'd like to admit. The PDF viewer in Internet Explorer appears to be much more developed and allowed digitally signing using the CAC card right in the browser window without needing to open up a local copy of Adobe Acrobat. 

Is it possible to drag and drop the entire document from within Adobe into a browser window? I noticed I can drag and drop the document tab into another adobe window but it doesn't allow dropping into the browser. This would greatly improve the user experience for getting their signed PDF forms back to the web server (then to the database). I've also thought about adding a button to the PDF form that would represent the entire document and which could be dragged onto the web page (similar to dragging a file from Windows Explorer into the web page).