For a 365 account the clean no install answer hides in Outlook on the web rather than the desktop client.
1
Open outlook.office.com and sign in with the same account.
2
Find the message, open it and click the three dot menu at the top right of the reading pane.
3
Choose Download. The browser saves the message as a real .eml with full headers and attachments inside.
4
Upload that file to the vendor portal.
Another zero install trick when webmail is blocked: create a new message in the desktop client and drag the original email from the list into its body. It attaches as an item you can then drag from the draft to your desktop, though on most Outlook versions that still lands as msg, so the web Download stays the reliable route. For recurring needs at volume, an MSG to EML converter processes a whole folder of dragged out msg files in one pass.