As far as I know, questions for the Twitter bot accounts are pre-selected. Mods have no control over what questions are pushed to the bots other than minor adjustments (The Twitter bot should not Tweet posts that have flags on them)
There used to be a form that could be used by moderators to promote questions to the bot by asking SE Inc., but I don't think it works again. The standard way would be to use your personal account to tweet posts.
Ultimately, we want to promote YOU, not the site itself. We want to be awesome by association with you. Asking you to subsume your entire Twitter identity into a faceless "Stack*" Twitter account does not strike me as the right way to accomplish this goal. - Jeff Atwood