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).
If you are looking for another solution, this is a gist on using the Mailgun service:
You need an account on Mailgun for it to work, but thats free up to 10k emails a month. You need to register a domain to send from too if you want to send off a whitelist.