Just as I'm about to argue the appeal in Gil v. Winn-Dixie in the 11th Circuit Court of Appeals this week, there is news on the Executive Branch's position on the question.
The US Department of Justice has responded to 103 Members of Congress and declared in a September 25, 2018 letter that there is no need for regulations to confirm that public accommodations' websites are covered by the ADA and must be accessible.
Whether that conclusion is supported / upheld by all the courts in the land remains to be seen.
Some circuit courts have held that all websites must be accessible; period. Some have held that websites which have a connection, a nexus, to a physical place must be accessible. Some circuits have not yet spoken to the issue.
The DOJ appears comfortable that the ADA statute itself is clear enough.
Anyone contemplating this question would be well advised to check with an attorney or someone else expert on the issue for advice.
The blog below is from Attorney Lainey Feingold's excellent blog, which I recommend. https://www.lflegal.com/2018/09/doj-cut/
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