Well yes, although the question was strictly about having an alt tag on a background image. IOW, something that would be visible if the image 404'd, or if images were disabled in the browser, which is not possible AFAIK.
Having non-visible text in most HTML elements is easy enough. There are umpteen ways of doing that. You can even just use color: rgba(0,0,0,0); if you like. That'll work.