I just built a similar feature into our app, where our users may specify email contacts and send chat messages to them. Those messages will be sent using a special email address containing our user ID, so we can route responses back into our chat database. By storing the email message IDs along with the messages, we can even implement full conversations and let our users and external recipients respond to specific messages.
This was rather cheaply possible using Mailgun webhooks - we even have delivery receipts!
Edit: in case somebody worries - this is a B2B application where buyers can talk to their suppliers, some of which may not be onboarded yet. We rate limit sending out too, so there’s not really a spam risk :)
EmailEngine allows you to build similar things on top of regular email accounts, so you would not have to use external services. EmailEngine would detect incoming emails from the account's IMAP server and send emails through that account's SMTP server + make it look like this was done using a regular email client (sent emails are uploaded to Sent Mail folder, replied messages get the \Answered flag in IMAP etc.
Be really careful with this kind of emailing flow. We were booted off 2 mains email provider because our B2B apps allowed people to send invitation to their colleagues... I'm thinking that big email providers (like Mailchimp) don't even try to understand your usage and largely prefer to ban any non-big players from their network.
Edit: in case somebody worries - this is a B2B application where buyers can talk to their suppliers, some of which may not be onboarded yet. We rate limit sending out too, so there’s not really a spam risk :)