You need to enable relaying for the internal network by editing a couple files in your /etc/mail directory....
root@alice:/etc/mail# cat access
And since it's come up once before, that "x" in there is me hiding my subnet. It is not actually an "x", and in your case, it would be a zero.
192.168.0.1 is the firewall. That line is there to allow connections, but to reject relaying. This is because I have a rule on the firewall that relays port 25 to my SMTP server to allow incoming mail to be sent to me directly. So. 192.168.0.1 is the firewall, the rule says to allow connections but no relay.
192.168.0 (no trailing period) specifies the subnet, when not already defined. The rule "RELAY" means that for all hosts defined by this set, allow relays. That's the internal network.
Once you've set up the "access" file, run "make". (there's a Makefile in the directory). That'll compile the access.db file that Sendmail actually uses. Once that's done, restart Sendmail, and it should work.