Many sites now have information on web accessibility, for example: TechDis. 

It is best to design accessibility of a website from the start; see WebAIM .

You can use any screen reader to test the accessibility of different websites.  If a site you want to use is not accessible, contact the website administrator and tell them what you need to access, and how access other sites.  

Here are two free screen readers: 

Webbie WebbIE is a web browser for blind and visually-impaired people, especially those using screen readers, used since 2001 all over the world.  It exists in 6 languages and can be translated into others.

Thunder is a free screen reader, available to individuals and organisations