Yes, an MX record is part of your organization's DNS setup. It's not part of your computer's setup: DNS records are what tell other computers how to reach your computer, and the MX record is used when they want to send you email.
When a computer sends an email to
firstname.lastname@example.org, it looks up the MX record for
somewhere.example.com. You can look up the MX records for a host name with commands like
dig -t MX somewhere.example.com or
host -t MX somewhere.example.com (and the
host command with no option also lists MX records, with the wording “somewhere.example.com mail is handled by …”).
If you only want to be able to send email inside your LAN, you need to set up an MX record on the DNS server from your LAN. That's probably the same machine that you've configured as a DNS server in
/etc/resolv.conf. These are technically two different functions, one is a DNS server returning information from its own database and the other is really a DNS proxy that forwards queries to the appropriate destination, but they are usually provided together. Contact your DNS server's administrator, or if that's you and you don't know what to do, tell us what you're using for a DNS server.
If you want to send email from elsewhere, you need a public-facing MX record. Most DNS providers, even cheap ones, let you set up MX records in addition to A records (machine names).