I thought maybe this is the issue: From Send emails from ERPNext from own account - #2 by umair by @umair :
To allow Users to send emails from your ERPNext account, you should ensure that Roles assigned to those Users also cover Email permission on that Doctype.
Update: It seems this is not the issue, as the user:
- tries to send Purchase Invoice document
- has roles of Accounts User (which allows Email-ing Purchase Invoice) and also Purchase Manager
If the “Email” permission is not given in the role, then the “Email” menu item will not be shown in the first place.