I doubt wireshark will be very helpful. Wireshark shows the network packet data. I'm certain the issue isn't the internet/network connection between the user and the website. The data (e-mail message content) gets transmitted to the website, and it just isn't getting saved into the message database. Unfortunately the website was developed in Cobol, excuse me, Coldfusion, so there's not a lot of developer resources anymore.
Based upon how many times we see little data hiccups, and then the page works on the 2nd (or 3rd) try, it sure seems like there is a database connectivity issue between the webserver and the data. Not enough sockets, buffers, bandwidth or something to handle the amount of traffic between the website and the data storage. It would be nice if the failure(s) would at least barf up a meaningful error message.
From this chair, I can only speculate. But I've seen this kind of stuff before in my adventures in IT.

