That's a new one on me. Generally speaking, while you can send keys to any number of screens and applications, it has to do with how that application interprets that "Keys" method. Essentially, TestComplete "hooks" into the object and the low-level Windows keyboard events to send those commands. Many times, those kinds of console windows have their own filters and hooks (like, to send CTRL-ALT-DELETE you need to use a different set of keystrokes, etc). So, I'm guessing that's what you're running into, something in the intercept of vSphere console is messing with those keystrokes.
Have you tried opening it via a standard RDP connection rather than vSphere?
Also, while I understand not putting the full TC on that remote box, is there any problem with putting TestExecute on there?