Microsoft is extending the HTML5 web client. Today (May 2021), there is a new function to allow uploading files directly into the session. The user can select the feature while starting a connection. Inside the HTML5 client, the user can upload a file with the icon in the upper right corner. The file is saved into the virtual drive: \\tsclient\Remote Desktop Virtual Drive\Uploads (the drive only exist in the current session)
An administrator can avoid storing files into the virtual drive setting the group policy (per user or per machine) to “Enabled”:
Administrative Templates\Windows Components\Remote Desktop Services\Remote Desktop Session\Device and Resource\Do not allow drive redirection