Is a Windows Server (e.g. Windows Server 2008) more vulnerable during the shutdown process compared to the Server is up and running? Does the process of shutting down Windows introduce additional vulnerabilities concerning password dumps/extractions?
In most scenarios, no.
In the normal working fine scenario, the processes that could cause sensitive data leaks are ended before the general system processes, so there is no danger here.
In the case one such sensitive process is stuck, the OS does not end task the others before action is taken for the stuck one (in this case the well known screen with wait or end task appears).
In the case of operating system getting stuck, there can be indeed a problem depending how complete that halt is, but this should not happen on W2008, unless there's some bad hardware involved or really bad software installed (in which case the system is already compromised).