"What do they want?"

This is the standard way to refer to an unknown person in English, anything else sounds awkward. "What does the person want", "what does he or she want", "what does this 'someone' want", none of these will sound natural to a native speaker.

There's not a man I meet but doth salute me As if I were their well-acquainted friend

English has always used the singular they, especially (but not always) when the gender of the target of the pronoun is unknown.