Absolutely, yet they're still decades behind the capabilities that SK or Japan developed domestically over the past 20-30 years.
Germany is not in a position today to start a nuclear program, becuase any nuclear program can be viewed as causus belli for a conventional war, and Germany would not have the ability to develop credible secondary strike capabilities if such a war were to happen in the next 3-5 years.
In addition, Germany lacks a domestic civilian nuclear program due to denuclearization in the 2000s and 2010s - something which Japan or SK DID NOT do.
Imo, it will will take 7-12 years for Germany to reach a point where it's domestic space/ballistics industry is comparable to even Iran's today.
That said, that development and capital might be sped up if defense related opportunities with the KSA are enhanced, because it would provide a massive infusion of capital.
Also, nuclear weapons programs are expensive, and Germany can better use it's capital to build conventional warfighting capabilities that it severely lacks, while remaining assured of France or the UK's nuclear umbrella.
Germany should have started this spending spree 10-15 years ago (which is when most other regional powers like Japan, SK, KSA, UAE, India, Pakistan, etc began doing so), but better late than never.
Germany is severely lagging in dual use technology, and German civilians need to realize that (the leadership does recognize this and hence why a spree of defense manufacting deals with the KSA, UAE, and India began during the Scholz chancellorship). That is the only way Germany can actually build strategic autonomy.