Not only can it be argued, that is the basic argument that has been deployed in similar situations. For example. race-restrictive covenants that forbade property owners from selling to blacks and other minorities are private agreements, but courts have refused to enforce them on the basis that the enforcement itself is state action.
On the other hand, Kleargear doesn't appear to have used the judicial system to enforce their claims, so the situation is more complex. The individuals could probably get the court to say that enforcement of the clause is unconstitutional, but I'm not sure that's what they're looking for.