You absolutely can host your own mail server. Or use one of your IT partner of choice. Out business still has an exchange. Today you have to jump through some hurdles with SPIF/DKIM, but it is still very possible.
You can also send out your emails via a relay, like Amazon SES, right? Host an email server (like opensmtd + Dovecot, or with Postfix+Dovecot), and forward external emails through SES.