The Joomla! CMS is not a mass mailing utility and should not be used as such. Though there is a Mass Mail component in the Joomla! core, Arvind Chauhan of JoomlaArt points out that "It’s not ethical [or legal] to send promotional emails to users who have not opted-in to receive them." The Mass Mail component does not have this opt-in feature.
As @Anibal mentions, ACYmailing is the most common and integrated solution for Joomla! for doing such. And, even though it is specific to Joomla!, it rivals the best 3rd party eNewsletter platforms available.
3rd Party Email Marketing Platforms
For a more universal approach and one that will work regardless of whether your on Joomla! or not, you can choose any number of email marketing services. Since you are a non-profit looking to pinch every penny, there are several services that offer free non-profit accounts:
If you already have a method for newsletter opt-in handling and simply need a beefy SMTP server for sending the emails, these services are free to non-profits:
In-House Email Marketing Platform
If you already have a method for newsletter opt-in handling and you believe your current SMTP server is sufficient to handle the emailing duties, won't get your hosting account closed, won't get the server blacklisted, or won't cause messages to be flagged as SPAM, but you need a beefier app to handle the creation and management of your email marketing campaigns, try G-Lock EasyMail7. It is free to non-profits.
With G-Lock EasyMail7 you can use your own hosts email system (SMTP) for FREE or enhance delivery with third party senders like AMAZON SES, that allow you to pay for what you use… Imagine being able to send 10,000 emails for just $1.