Robert Jerin is right with WW I, but I think war began in summer (August?) so there *was* a chance to travel before that date ...
(with it's English language option) might help you further since you don't seem to have a detailed address, just Villach. At that time any vital records were kept with the Catholic church only, meaining you must know the parish.
Anyhow, there are about 14 Warzilek listed, a few of them in Carinthia. Maybe writing a few letters would halp.