This depends on how your scripts are delivering the messages to the local server. If you're connecting to it over TCP/IP and transmitting the message that way you shouldn't have a problem, just change the delivery host in your scripts/configuration to the external server (and make sure you've configured that server to only relay mails from your own machines!).
If your scripts pass messages to the mail server on the command line this won't work, they will still require a local SMTP-capable mailer. This doesn't have to be a full-fledged sendmail installation though, you could write a script which receives mail on the command line and delivers it to the smarthost, or use one of the minimalistic sendmail replacements that exist (e.g. femail or mini-sendmail, search freshmeat for more).
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