That is a function that can be enabled/disabled on the result list dialog, at least in 6.11 and prior. Not sure if that changed in 6.12. So you'd need to make a separate Role for the users you want to be able to change multiple documents at the same time, with a specific Search/Result dialog combo assigned that has the function enabled.
I think what you're looking for would be a functional profile. User Administration - Functional Profiles in the admin client. They can be assigned to a ROLE. Once upon a time in the thick client days, we could control what options showed on each and every dropdown menu - FIle, Edit, etc with a functional profile. I think they show up in DWGeneralProfile in DWSYSTEM. You can assign various stamps the same way, if you only want a specific group to have access to a stamp that updates a certain set of fields - like a user audit/QC type process - just make a profile for "audit stamps" and assign that profile to only the role(s) you want to perform that funciton.
Document trays are essentially a file cabinet, so all their settings will show up in the file cabinet tables in DWSYSTEM. It takes a bit to piece it all together but you should be able to see what users (DWFCProfileToUser or DWFCProfileToRole) are assigned to a particular profile (DWFCProfile). We don't use C2O, but C2M configs show up in DWUserSettings.
I know I've come across the Import Configs somewhere, but I can't find them right now!