Another federal agency, the Architectural and Transportation Barriers Compliance Board (ATBCB), also known as the Access Board, issues guidelines to ensure that buildings, facilities and transit vehicles are accessible to people with disabilities. The Guidelines & Standards issued under the ADA and other laws establish design requirements for the construction and alteration of facilities. These standards apply to places of public accommodation, commercial facilities, and state and local government facilities.
Level AA is a little more significant, and makes sites accessible to people with a wider range of disabilities, including the most common barriers to use. It won't impact the look and feel of the site as much as Level AAA compliance, though it does include guidance on color contrast and error identification. Most businesses should be aiming for Level AA conformity, and it appears to reflect the level of accessibility the DOJ expects. 
The type of operating system and hosting doesn’t matter much for design, but the platform is a factor. For instance, if you need a WordPress theme, the designer must work with WordPress themes and customize them in a way that a developer can feasibly code in CSS or JavaScript on the front end. If you use another CMS, the developer must work with those elements. Most designers specialize in specific platforms, so review their portfolios and experience for specific platforms before you invite them to bid on your project.
Most recently, however, pizza chain Domino's has been brought under suit for their website not being accessible for specialty ordering. The U.S. Supreme Court refused to review the case, instead upholding the decision of the 9th U.S. Circuit Court of Appeals who said the “alleged inaccessibility of Domino’s website and app impedes access to the goods and services of its physical pizza franchises—which are places of public accommodation.”
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