Using any provides emphasis. In this example, the emphasis occurs in two ways.
First, more than one customer may have expected a response, but not one of them has received a response. The emphasis is that out of all of those expecting a response, none received one. This is a stronger claim than saying that only a few received a response, or leaving that possibility open for interpretation, for example.
Second, it can be interpreted to mean that no customer received a response in any possible form. For example, no one received an email, a letter, a text, a phone call, or a smoke signal. This is stronger than simply saying the airline didn't respond. It claims that the customers might have found any type of response acceptable, yet no method was used whatsoever.
Of these two cases, determining if one or the other cases was intended can't be determined by the sentence alone. But it can be said that, intention aside, the words support the claim that both cases are true.
Replacing any with a might have the same literal meaning, but the emphasis is lost.