You should definitely just fix this at a system-level.
DO NOT try to work around sysadmin errors in software. Get your systems admins to set servers up in a way which makes life easy for you - that's what they're there for.
And NogDog, no you can't simply run sendmail from the other machine's drive, that wouldn't work.
I find it hard to believe that any production Unix box has no MTA at all installed, as other unix facilities use it to mail error logs etc.
Mark