Welcome to the Wefugees Community and thank you for your question.
People who hold an "Aufenthaltsgestattung" (still in the asylum procedure) need the permission of the immigration office (Ausländerbehörde) to start the vocational training ("Ausbildung"), which can normally be applied for after three months if you don't have the obligation to stay in a reception camp. As far as I know, there is a right to get that permit after 9 months of the asylum procedure - even if you are still in a reception camp. Unfortunately, this is not the case for people from so-called safe countries of origin who applied for asylum after the 31. August 2015.
People holding a "Duldung" need the (work) permission of the immigration office to start vocational training (Ausbildung). Those who don't live in reception camps can apply for it immediately. People who stay in reception camps might be able to apply after 6 months. However, as there is the opportunity for a special Duldung, the so-called "Ausbildungsduldung", there could be an option to apply/start to work before the end of that 6 months. There is even a right to get the "Ausbildungsduldung" if you fulfill all the requirements for it. At the end I would always recommend to apply for the work permit - especially when there is a concrete job/training offer already.
Please see this recently published brochure for details (in German though/esp. page 33).
I will also link two of our wonderful experts here. Dear @mbeon-Christine and @mbeon-Ruth, would you agree with my explanations and/or is there anything you may want to add?
In any case, please don't hesitate to get back to us with any further questions.
All the best,