A hypothetical Chrome Inc. (or Chrome Foundation) could make its money off of selling rights to the search bar (hypothetically to Google anyway), and, I suppose, donations. Enterprise support would be another thing they'd have to provide (and could sell), especially if they got the Chrome OS project.

It'd be a speed bump, and they wouldn't have the full resources of Google, but I think it could make it. (But I don't know if it should, the open-source Chromium base isn't committed to only by Google.)