I work for a company that makes iOS and Mac software for people with physical, vision, communication, and reading impairments. Primarily I work on AAC (Augmentative Alternative Communication) apps, but I have a broader interest in Accessibility (I just returned from CSUN in San Diego, which is more for digital accessibility than web).
A fantastic free course on Digital Accessibility can be found on Future Learn: https://www.futurelearn.com/courses/digital-accessibility
With regards to performing a web accessibility review, I have done a few of these as a freelancer - my process is:
1) Run the site through WAVE (http://wave.webaim.org/) - or an automated checker of your choice.
2) Run it through a readability checker to check the Flesch-Kincaid score
3) Tab around the screen to make sure that the focus is shown, and nothing funny happens!
4) Use a screen reader to make sure semantically it is correct, and nothing is missed (ALT text - poor heading structure - etc.) - I generally use JAWS, but you could use NVDA or VoiceOver.
5) Go through the site from a UX lens to try and cover everything else that automatic / manual testing don't pull up (eg - maps / videos with captions)
6) I then write the report with ways to fix the problems, color, text, code, etc.
7) FINALLY - I then strongly advise that the company reaches out to people with disabilities and perform usability testing.