Hi,
For a better troubleshooting, I suggest you can do as the followings:
1. As the user can upload document using Ribbon menu, It seems this is not the permission issue, I suggest you can check if the Internet Explorer Adds-On"SharePoint DragUpload Contorl" is enabled.
Here is a detailed article for your reference:
Fix for SharePoint 2013 Drag and Drop Upload Not Working Issue
2. You can try to create another document library and test if the contribute permission user can drop and drap to upload file.
Thanks
Best Regards
TechNet Community Support
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