It worked for me with my existing Amazon account, but then I have a Kindle.

I thought I'd test with a new account, so in an incognito window:

  * Went to https://read.amazon.com/
  * Created a new account
  * Got redirected to Kindle store, found free book, hit "Buy now with 1-Click"
  * Filled in a billing address
  * Got a page thanking me for my order, hit "Read now in Kindle Cloud Reader"
  * Book loaded

You're welcome to use this account--login: "M8R-2neeuu@mailinator.com", password "test123".

Hope this helps.