No, a running program can't be uninstalled, so it's impossible a program uninstalls itself.
The "legit" site you mention only offers me a login screen, so is no help at all.
And the logs only show you run their chat program from your appdata\roaming folder.

However, they are the only ones that can help you with your issue. So either try again or accept it's a program that isn't suitable for you.