I found a comparison of e-mail service providers that might help you find what you are looking for - it has a neat guide that can give you an idea of what's available. Since cost is an issue, I would look for services that charge based on your subscriber count, as opposed to the number of e-mails you send.
I can't say, one way or the other, that any of these companies would be willing to offer their services to you for free - but it couldn't hurt to contact them and ask if they have special programs for non-profits. If the mission of your non-profit resonates with someone in one of these companies, you may find yourself with a sponsor.
You may also want to look for technical consultants/contractors that might be willing to offer their services for barter with your organization to help you build your own crm/mailing system - along with all the technicalities and training so you don't end up getting your mail server blacklisted left and right. It really doesn't take much to do this yourself, provided you do have some technology resources and something to barter with that you can leverage.
Along the same line, if you do need computers and technology, take a look at various recycling centers. They often donate computers that were discarded by other companies to non-profits. With a little poking around, it isn't very difficult at all for you to find enough hardware to run a half-decent mail server that can run some open-source software.
The key here is communication. Call around and talk to people. You won't find much of anything being advertised on the internet, nor will you get much from a group of folks trying to build a profitable startup. ;-)