You asked: "Would the next step be the war archives in Vienna? If so,"do" you know if they will answer an email inquiry or is it best to write a letter?"
Maybe. The Austrian Archives have the WWI records for Austrian soldiers. But it is unlikely that they will have any records for your area of the Empire for WWI. They would have records for all officers but not for the enlisted men. WWI records for soldiers from most regions outside of the current boundaries of Austria were recorded by or given to the successor nations. Sadly, most of these records were destroyed.
The Kriegs Archive will respond to emails but to do a search they will ask for a reasonable fee to do it. I always hate to say that finding the military records for a given individual is unlikely but I do think that is the situation here. Sorry.
Have you thought about looking for records of earlier ancestors? They are much more complete.