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You can log all the output from a script by sticking something like
exec >/tmp/scriptlog 2>&1
at the top.
This doesn't rely on having an MTA installed.
In general it'd be better to have an MTA for cron to use, even if it's a minimal one with local-only delivery, like this.
The only problem with a logfile option like that is that it won't log out problems with the script itself, those will only go to stdout/stderr and if cron can't find an MTA to send them on via email they're just being lost as they don't get logged anywhere else.