No, you do not need an MX record. Mailservers can function without.
But in general setting up an internal DNS server beats the alternative: maintaining hosts files and/or using IP-addresses everywhere.
Generally sending email messages to user@internal.example.com
will also come easier than using the square bracket syntax to send messages to a bare IP-address user@[10.9.8.1]
.
Normally an MX
record is needed more by other (SMTP) servers to discover what server handles the incoming e-mail for a particular domain than for your own systems. You have full control over your own systems and they don't necessarily need such discovery mechanism, they can be explicitly configured with the hostname/IP-address of the host they should deliver their messages to. So an MX record is not mandatory and you can theoretically do without. But when you're setting up internal DNS anyway, adding that MX record is a trivial effort.