The civil rights acts covers, at most, race (ethnicity), religion (requiring reasonable accomodation at no more the de minimis expense), gender and handicap. The common law might require admitting customers who are behaiving in an orderly manner, but wouldn’t allow much in terms of damages.
The initial posting didn’t refer to anything, but it would be very hard to challenge a store over a dress code unless you could argue that it violated one’s religion or that it interfered with one’s handicap. IF the issue was employment rather than shopping, it might be a little bit different, for example, there have been lawsuits (not winning ones) objecting the dress codes. Stores routinely have dress codes.
I checked google and found complaints, none of which referred to lawsuits, and one has someone at Yeshiva University’s law school (which is part of the non-frum part of YU) complaining about the inability to sue over a dress code, and an article in a less than reputable New York newspaper showing some non-Jewish women dressed in a manner that almost anyone would consider more than appropriate for a “lady of the night” than a customer in conservative family-oriented store.
Edited.