Be Zen about it and send them an email stating that after x developer- hours and y tester-hours of trying, you as a group were unable to reproduce the problem. Note that this complaint is the first incident of this type that the group has been notified, after the customer's software has been deployed and in production for six months. cc: your management and the same people that the customer cc'ed. If you are not customer facing, have your manager send the email.
The ball is in their court, and the burden of proof is on them. Your story reminds me of a particularly ignorant sys admin who called our tech support, stating that our load balancing software was no longer working on one of his machines. The field engineer we dispatched established that the sys admin had moved the machine to a different location within the office, without taking note that the new machine was now on a different IP subnet and without paying attention to the fact that the machine had been assigned a static IP.
Until new info surfaces, my best guess is that your customer did something stupid on that day. My second best guess is that there was a network glitch that day. Third possibility is that some of the data your software was processing was corrupt.
Keep in mind before you go to war that the world's idiots have us significantly outnumbered, and that they'll outlive you and me - While idiots may aggravate us from time to time, they have to live with their idiocy 7x24. For them, that's what hell looks like :) There is not much that you can do to them that's worse than what they're doing to themselves and to each other.
@jcm comments that " may come off to the customer as "It works on my machine" which is never acceptable to customers or bosses. It might be helpful to put in the email a request for details from the customer to help you reproduce the problem. That way, you're seen as cooperative while demonstrating that the problem isn't on your end."
That's NOT a good suggestion! I am presuming that the OP had the good sense to ask for these details beforehand from the customer when they tried to reproduce the problem. It makes no sense at all to try to reproduce ANY problem without data - This is so fundamental as due diligence that your suggestion would come off to me as somewhere between not funny and extremely frustrating if I were the target of that suggestion.
I couldn't care less about being viewed as cooperative. The only thing I care about is reproducing the problem. I am not into perception engineering.
As I had said, the OP's team, despite their efforts, were unable to reproduce the problem, and the ball is in the customer's court. Especially when "It works on my machine", it works on every other customer's machine, it has worked on the customer's for the past six months and the only time it didn't work was that ONE instance.
Come to me with such a story that my software failed, and if not only I but my entire team can't reproduce the problem with the data that you gave me, I'll shrug off your story. If either you or my bosses were not happy with our efforts and you know something that we don't and you feel that you can do our jobs better than we can, you are all more than welcome to reproduce the problem by yourselves and show us up.
Clarification from OP: "We are responsible for the software working but it runs on IT infrastructure they are responsible for setting up and maintaining at every event. They literally have no one who is qualified to do this. They had one qualified network engineer that they made redundant because he used to sit around not doing anything because they never had any network problems ..."
They could have had the qualified network engineer set up for every event. They did not come exactly ahead when things did not go according to plan at that last presentation.