We have several external websites in which we need to upload documents. When the external website's upload tool opens, you are presenting with File Explorer. There doesn't seem to be an easy way to map to SharePoint Online to get the document. The only work around we've found is to download the file to our desktop and then upload to the external site. Definitely more steps than what it should be and we run the risk of not deleting the downloaded file.
@nwhitacre You may have solved this by now but the traditional way would be to sync the document library or folder where you have files you want to upload to the external website.
We have a problem where certain data cannot, by policy, be on the desktop or laptop, and sometimes those files have to be uploaded. It looks like Edge may be able to open document libraries in file explorer with a registry edit and tenant parameter change.