What open access journals that publish research in cognitive science/psychology exist?
Apart from PLOS ONE, are there any other open access journals that publish research in cognitive science/psychology?
Where should I start looking for them?
The Directory of Open Access Journals is a great place to start when searching for Open Access Journals in any field.
You can browse through journals for the specific subject areas like Psychology or Neurology, or you can search for journals or articles containing certain keywords.
The DOAJ lists articles in multiple languages as well, not just English journals.
Another resource is Sherpa/Romeo which aims to be a site to help you "find a summary of permissions that are normally given as part of each publisher's copyright transfer agreement".
For example, here's what it says about Journal of Applied Psychology.
That said, I'd always double check the accuracy of the information on the site with the actual journal home page and the actual publishing agreement that you sign.
One thing that hasn't yet been mentioned: if you find a paper you want that's not open access, many researchers nowadays put pdfs of their work on their personal sites. The easy solution is therefore to google the name of each author, starting with the first, plus something like "Publications", or if they have a common name, add "Publications, Psychology" to the search terms. Google scholar seems to often be able to do this and help you hunt down the papers easily as well.
Finally, let's not forget that you can just email the first author and they'll send it back to you.