My website is designed to meet the accessibility guidelines.
I'm HOPING that this means screen readers should work well with it...... But I have two questions
NVDA is a free option:
NonVisual Desktop Access (NVDA) is a free and open source screen reader for the Microsoft Windows operating system. Providing feedback via synthetic speech and Braille, it enables blind or vision impaired people to access computers running Windows for no more cost than a sighted person. Major features include support for over 20 languages and the ability to run entirely from a USB drive with no installation.
Is this a fair assumption to make?
No, even if you think you know the ins and outs of accessibility well, only testing can really tell you this.
Are there any free/cheap screen readers clients I can use to test or online emulators?
The already mentioned NVDA looks like a viable option, or you can download a trial of JAWS which I believe is the most widely used screen reader on Windows. If you're really serious about accessibility and you have a requirement for ongoing testing, you might want to think about just buying a copy.
On a final note, it sounds like you already know this but no amount of automated accessibility testing can really tell you if your site is accessible, only real-world testing can do that! Which is what you're doing, so well done.
Just because something meets the guideline doesn't mean it's guaranteed to be accessible, all screen readers have there different quirks. I'm a totally blind individual so comments on screen readers are below.
Note this is a rather long post so I’ve summarized it at the top. In summary if you want to make sure your site is mostly accessible use NVDA, if you want to make sure that blind individuals working in the government will be able to use your site use Jaws to test, if you want to be extra safe use Window-Eyes and Orca to test as well.
NVDA is an open source screen reader that is rather new. It isn't quite as good as some of the commercial screen readers out there but it gets the job done. I'd say if a site works with NVDA it's likely to work with most other screen readers. One issue with NVDA is the fact that its accessibility is only really good in Firefox so you'll have to use that to test.
Jaws is the most widely used screen reader out there. You can download a demo of it that will run for 40 minutes at a time then require you to reboot if you want to run it again. If you’re trying to insure 508 compliance this is probably the way to go since Jaws is the screen reader used by the US government.
Window-Eyes is the second most used screen reader. I don’t have any experience with it but I’ve been told it’s quite good as far as internet accessibility goes. Orca is a screen reader built into gnome that works with Firefox and Linux. It’s built into Ubuntu. I tried it about a year and a half ago and it was absolutely horrible but I’ve been told it’s gotten better.