Apologies for the delayed response here! I didn't get notification of your response for some reason!
Glad to hear you have something (almost) working! So typically sending emails relies on an external entity that relays emails. For instance, inside of a data-center, a company would have their own SMTP server that relays all emails (i.e. smtp.mycompany.com). The idea here is that you would point Connect at that SMTP server with the appropriate credentials so that it can then be used to relay emails from Connect to your inbox.
Some cloud providers have such SMTP services, and there are a handful of external entities out there (mailchimp, etc.). On AWS, I am familiar with "SES" (Simple Email Service), which can be used for relaying emails over SMTP. I'm not sure if/what DigitalOcean offers in that realm.
What is the scope of your proof of concept? Does your organization have an SMTP server? If you're expecting this thing to live for a while, you could either look into whether Digital Ocean offers any such service or if a third party provider makes sense. The other thing you could do is look into relaying emails on behalf of a user in a more standard mail setup like Gmail (although that comes with caveats). This article seems to do a decent job of exploring the tradeoffs, and I have no idea what biases they may have 
Your best bet is probably to find out what standards your organization has for sending programmatic email and then see if there is a nice way to piggy back off of that 
If you're cloud-agnostic and AWS is an option, we do have a cloudformtion template that can get you the whole of RStudio Team setup in a pretty basic setup with just a handful of clicks:
https://docs.rstudio.com/rstudio-team/cloudformation/
You would still be required to setup SES yourself to relay emails, but I know that is an avenue for cloud email delivery with a cloud provider, and there are lots of tutorials out there on the topic (which is how I learned) 