Theaters often wait until the last minute to cast. That’s usually May or June, but sometimes they even wait until the beginning of the Season (Spielzeit) in September and are desperate ! So while it can be early in the game for most,, September can be prime time for getting a contract, for a production (Stückvertrag). October and November used to be prime time for theaters to find new ensemble members (Festverrag, or Vakanz), since both singer and theater have to give notice by the end of October that they intend to leave or not rehire.. This is still true, but there are far fewer of these salaried positions now, and this time slot isn’t what it used to be.
The two or three weeks before Christmas and Easter are dead zones for theater auditions, which is convenient since airfares are high at these times. As for January through pre Easter and post Easter through May/June, if theaters are looking for you it can be a great time, but it’ll be short notice and you have to be there to respond..