If you want to allow relaying by IP, sendmail reads a file named /etc/mail/access. I will assume nothing fancy like an LDAP backend. I'll assume that your local network is behind some kind of masquerading firewall, such as a Linksys. I'll assume that your local network is 192.168.1.0/255.255.255.0.
Edit the file /etc/mail/access. Add to the end of the file on a single line by itself:
Connect:192.168.1 RELAY
If you have other local networks that you want to allow relaying for, you can put them in here as well, one per line.
After you have saved this file, recreate the access.db file but changing into the /etc/mail directory and run (as user root) 'make'. There is a Makefile that RedHat provides that rebuilds the individual *.db files in that directory if it's needed.
When you rebuild the *.db files, you do not need to restart sendmail. You only need to restart sendmail if you edit the /etc/mail/aliases file or create a new sendmail.cf. It reads both of those files once at initial startup, the rest of the *.db files are reread for each incoming email that sendmail processes.