“Incompetent” is a strong word - probably too strong. I think it is fair to say however that the employee is not maximally competent. That is, there are certainly people out there who would have handled the situation better than your employee, though perhaps such people are not easy to find and recruit for the type of position your employee has (or indeed, possibly, for any position).
Let’s examine how your employee behaved and how a more thoughtful person might have handled it.
The employee said our system does not accept clients with no mobile phone numbers, so she would keep the old number in place.
A maximally competent employee would be sensitive to clients’ concerns about large corporations collecting data about them, including things like old addresses, phone numbers etc. Even if it is true that the system requires a phone number, a better response would have been to tell the client “I have deleted your old phone number from our system. However, the system does not allow me to leave the phone number blank, so I’ve entered the number 000-000-0000 instead. Be assured that this won’t be an issue in any practical sense.” If I were the client, I’d be much happier with such a response.
The employee said the client's tax ID number wasn't in the screenshot, and the client couldn't remember it offhand either. The client asked if the employee had access to the internet so he can find the number. She said no - our computers are only connected to the intranet. The client said fine and left.
All the talk of IT security notwithstanding, sending a client away empty-handed because of something as trivial (in 2019) as lack of internet access seems shortsighted to me. Again, even if the reality is that the branch’s computers are only connected to the intranet, the employee didn’t even attempt to think creatively or even show the client that she would like to help. She just said no. She could have said “oh, I can set up a hot spot with my phone, it’s easy”. She could have said “give me a minute, I’ll consult a colleague”. She could have done any number of things to show the client she cared. Even if she had failed and the client still had to leave and come back, I bet they would have ended up a much happier client.
One hour later, the client came back with more screenshots and the ID number. The employee said she needed the documents printed out.
Well, it was a clear mistake on the employee’s part not to mention the part about needing a hard copy. As a single incident it doesn’t say much, but an accumulation of similar mistakes would truly be evidence of at least mild incompetence. In any event, the client’s frustrated reaction is completely understandable and predictable.
Even after the mistake had been made, the employee could have (and should have) gone out of her way to help the client print out the screenshots. Sorry, I don’t buy the nonsense cop-outs about IT security. How about using a camera (on your phone, or one of the branch’s computers) to take a picture of the screenshot and print it out? How about helping the client finish whatever business they had and telling them they have to email the screenshot to the branch later? How about just acting in a way that gives the impression she cared? All of those are things that a maximally competent employee would likely have thought of doing, and that would probably have changed the outcome of the story.
The client "threw his arms up in exasperation" and said, in that case, he would like to close the account. The employee closed the account, and the client walked out
Again, the employee’s response shows lack of caring, empathy, and imagination. She had an opportunity to stop the client from closing the account using some combination of creative thinking, charm, apologies, consulting with colleagues and whatnot. She didn’t even try. She may not be incompetent, but it is a certainty that some people would have handled the situation better.
Finally, I do not mean to suggest that the above analysis absolves you and your company of any blame. Just like the employee is not maximally competent, your procedures also seem not maximally competent, and it seems likely that the training your employees are given is also somewhat at fault. Good training and good procedures will squeeze more competent performance even out of mediocre and mildly incompetent employees. With that said, one cannot have procedures that cover every possible scenario (in particular, I completely disagree with the statement “If you don’t have documented procedures for what to do if a client brings in screenshots [...], you have way bigger problems than one employee” from @AffableAmbler’s answer), so it’s reasonable to expect employees to be able to think on their feet and have some measure of common sense when some unusual situation occasionally arises, as it inevitably will. If what you really meant to ask was “am I reasonable in being somewhat dissatisfied with the way the employee handled this situation?” my answer would be “yes”.