I'd like to set up alerts that send notifications to email addresses that do not belong to any members in my Sentry org.

It is currently not possible

There is an open feature request to add this feature, which you can follow here.
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The current workaround is to add a member to the organisation with the email address you want to send alerts to, and configure your alerts to send an email to this user. If email isn’t strictly necessary, you can also use integrations like Slack or Microsoft Teams to automatically route alerts into your existing support channels.