Learn English – Question starting with OR

Of what audience?

Scholastic English traditionally regards starting a sentence with a coordinating conjunction to be an error.  For that audience, a compound sentence is more acceptable:

Maybe we can do something about it, or is that impossible? 

You might know this general pattern by the name tag question.  If not, you may still recognize it as a compound sentence.  Despite the fact that the question no longer forms a sentence on its own, it remains an independent clause and the sentiment remains much the same. 

Or, does it? 

In conversational registers, it's common.  For example, I see such questions standing not merely as separate sentences but as independent paragraphs in my professional e-mail correspondence.  I've used it myself, just as I've used it above, to mark a sharp transition in the body of the text. As a tag, that question cannot serve that purpose quite so clearly. 

Conversational registers are far more lenient regarding sentence fragments than formal or scholastic writing.  For example, the first line of this answer is a prepositional phrase.  It's a grammatically sound prepositional phrase, but it certainly does not constitute a clause in formal writing.  The fragment makes sense here only because you provided the context that makes the question sensible: "Of what audience [do you want to ask such a question]?"

It's not impossible, it's not even unlikely, but it's not appropriate under every circumstance.