I do not know Azure or Sitecore, however, I did do some poking around and this is what I found.
While it appears that some sitecore cookies can be disabled such as analytics, I see no reference to disabling the cookies for login and other management aspects of the CMS.
As far as Azure is concerned, the only reference to what you are asking is for static sites.
And while I am not entirely sure what you are trying to achieve as far as your website is concerned, this is not clear, it appears that the use of Sitecore will require cookies.
On Use cookie-free domains, it appears the recommendation for cookie free domains is for statics sites.
You may not be able to have a cookie free domain using Sitecore.