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You should start with the Deutsche Nationalbibliothek.
The German National Library is entrusted with the task of collecting, permanently archiving, bibliographically classifying and making available to the general public all German and German-language publications from 1913, foreign publications about Germany, translations of German works, and the works of German-speaking emigrants published abroad between 1933 and 1945.
This is not a "retroactive" collection - the mission to collect all published German works actually started in 1913. Even in Nazi Germany a copy of each dissertation would have to be delivered to the Nationalbibliothek, so they should have a fairly complete set. And they have an online catalogue, so you do not have to travel to Berlin (although our fair city is always worth a visit).