In a quick search, it doesn't appear that there is a legal consensus that a website such as Chess.com would count as a "public accommodation" under law. From what I'm seeing, from an admittedly short search (and IANAL), is that there have been rulings that go both ways, as to whether or not a game site, like this, falls under the regulations.
The best method to voice your concern with the site would be to communicate directly with staff by opening a ticket:
https://support.chess.com/customer/portal/emails/new
So I have suggested this several times nicely and gotten nowhere. Nice is not necessarily one of my best things.
Chess.com has millions of participants and, in fact, loves touting their membership in 8 figures on their home page. There are currently only two ways to input moves into chess.com -using a mouse and using a touchpad. It is simple to envision more including keyboard entry, voice entry, or simply extensions that allow people to create whatever entry system they want - someone so inclined could use extensions to create a retinal tracking input application like on a gun ship. But those of us with hand problems have extreme difficulty entering moves on chess.com. It is very frustrating.
Chess.com is a public accommodation subject to Title III of the Americans with Disabilities Act. For awhile it was unclear whether a website was a public accommodation but now that defense doesn't work anymore (I'd be happy to give a list of cases). Chess.com has a much weaker case than just about anybody else trying that defense as memberships are free for the part that affects disabled people and they put their 10M+ membership number on the front page.
Chess.com has launched all kinds of other services including Chess960, ChessKids, and now 4-person chess (the one that tipped me over the edge). The defense that they don't have the resources to comply with the law is obviously false.
Vast numbers of elderly people have problems with their hands. Many younger people do as well from accidents, carpal tunnel syndrome, other diseases. The class affected is large. The lack of input flexibility affects other classes as well, notably blind people who are doing a decent job with these lawsuits.
So I am asking now less nicely for chess.com to do something about complying with the ADA. In the U.S., you can't ignore laws because you don't like them or somebody else ought to take care of it. Title III absolutely requires accommodation of handicapped people as a public accommodation. I would much rather have this go peacefully with changes made to the website to make this work for everyone. There are very large numbers of people who would like to use this accommodation but can't because your programmers are busy writing 4-person chess applications instead of complying with the law. Completely unacceptable. Let's save me the trouble of sending the certified letter this week, writing the complaint, filing it in Federal Court, all that messy stuff.
Thank-you.