While the sendmailR package should work fine in Shiny Server, it will definitely not work on shinyapps.io.
In more detail, shinyapps.io does not prevent users from sending e-mail using their own provider. Meaning, you can connect to MailChimp, Mailgun, or any other SMTP email service. However, we do not provide e-mail sending ourselves and the Unix sendmail program (which sendmailR wraps around) is not configured to leave the host it is running on, so you won't be able to use that (no matter which tier of shinyapps.io you have).
Hope this helps!