Software leaders: what’s the best way you’ve found to get to know key stakeholders when you are new to the organization? How do you build enough trust and rapport to get honest feedback from them on how to improve the department?

Getting face-to-face is the best way to connect.  The approach has to include connectedness to the person not just the role they play in the organization.
In a hybrid world that most likely will be in a virtual meeting with your cameras on.
Learn about 
- who they are
- what do they care about
- who are their people
- explore ways you can support them and make things easier for them.

Having one-on-one, face-to-face meetings is always a good way to build rapport.  Focusing on customer service and being open with information and knowledge builds trust. From a technology perspective, if you have a ticketing system, acknowledging of a ticket within the SLA is always a good way to let the stakeholders know you aware of their request.  Also lets stakeholders know you are committed to their sense of urgency and setting the proper timeline expectations is always helpful. In other words, take advantage of the organization's preferred communication channels and stay connected.