Just in case someone else meet the same problem.
I had the same problem about running Nodemailer in Azure Website. Finally I found the problem was that I forgot put Nodemailer as a dependency in package.json. Then azure did not know it had to install Nodemailer for the hosted server.
Add "nodemailer" : "*" at the end of dependency list in package.json. Don't forget to add a "," after the last line before you add "nodemailer".
Hope it helps.