Thanks for all your responses to my question on sendmail.  I got mail to
work and thought I would post my resolution for others that might have the
same question.  It turns out that I did not really have to change anything
with sendmail.cf even though I don't use DNS.  By addressing the mail
message to userID@[IP Address of internal SMTP gateway], I was able to send
mail anywhere in my organization.  The key was putting the IP address in
square brackets.  Sendmail interprets this as the IP address to send the
message instead of trying to lookup the address of a host called "IP
address".  I did make a change to my sendmail.cf file, but only to change
the way the "From" field looked to recipients.  I did this so people could
reply to my personal email ID instead of the root id of the netview box.

Example command to send mail message: echo Test | sendmail -f philipt
philipt@[10.1.2.3]

Changes to sendmail.cf:

DwNationwide
Dmcom
Cw Nationwide $w $?m$w.$m$.
Dj$w.$m