There is no simple answer to this.
The user can configure Outlook to send messages immediately, but this will apply to all messages that they compose. They may not want to do this.
I do not know of any way to send just the messages sent to Outlook from another application immediately.
Alternately, you could send mail directly to a SMTP server and they will go out immediately. The downside is the user may not have a SMTP server or know what it is and how to configure it. Also, they would not have a copy of these in their mail program.
Using Outlook, each person has a copy of the mails they sent in their copy of Outlook. But, nobody else has access to those records. So, ideally the app sending the mails should be keeping a log and copy of all outgoing emails.
No simple answer.
Regards,
James