In-page 3rd party plugins for reading websites aloud are no longer useful for websites, and I'd even go as far as to say they're worse for accessibility than not including it.
If people benefit from using screenreaders then it is highly likely that they would have discovered this long before visiting your website. After all, Jakob's Law of the Web User Experience states that "users spend most of their time on other websites" so, regardless of how important your website is, users will have been using the web elsewhere outside of your site before finding it, and as a result would have likely discovered or been made aware of assistive technologies such as screenreaders (JAWS being the most recognised paid-for system and NVDA being the most common free open-source alternative, not to mention Apple's VoiceOver which comes as standard with OSX).
So, provided your site is marked-up correctly to W3C and WCAG2.0 standards then these users who already use screenreaders will be able to navigate your site using the skills and techniques they already know from their screenreader of choice and wouldn't have to learn a new screenreader system just to browse your site (this is the 'worse for accessibility' part I mentioned in my opening paragraph).
What you can do though is to include a clear link to an accessibility information page on your site where you can provide information and links about the various screenreaders that are out there (there are more than just JAWS and NVDA) so if the user is interested in these and hasn't already come across them since they've been using the web then you are providing a useful service to them. And you'll be helping them to acquire and use assistive technologies that can be used on your site and on nearly every other site they visit around the web, so you're benefiting web users as a whole, rather than just users to your own site.
I would also add in as a response to your requirement that the site "needs to conform, as it's not for profit, to various online accessibility indicators" by stating that there is no requirement under WCAG2.0 recommendations that sites should include read-aloud systems embedded in the page. Not even if you wanted to qualify for AAA accessibility standards (which the vast majority of websites don't look to meet due to the incredibly complex overheads that come with having such a site). For example, one verified AAA site is http://visionaustralia.org/ and they don't have inbuilt screenreader technology. (AAA as verified by W3C themselves)