Online is typically faster. But not by much. My preference is to mail petitions in so I can make sure everything is there. Don't staple or tab anything as that does slow down processing by the contractor.
I'd find it hard to believe that he doesn't have at least enough interest or passive income that wouldn't trigger a tax return. But if he didn't need to file then at least a letter from him to explain that will be needed. Bank statements will be used to show the sponsorship ability.