Back in the olden days (more years ago than I care to consider), I was a management analyst doing efficiency studies for a large government agency. Every two or three weeks we went to another location and had to establish ourselves with a group of people we had never met before. I did this for 9 years. This is very similar to your situation.
First, you can get no cooperation at all without management cooperation. So first you need to go to the manager of the area you are looking at and give him a presentation about your group and what you hope to accomplish for him. Focus the presentation on what is in it for him. If you have senior management buy-in (and I presume you do or they wouldn't have created the group), make sure to mention who is supporting the effort.
Get the manager's OK to talk directly to the employees in the areas you are interested in. Tell them that this is their chance to fix the stuff that has been annoying them for years. Get them to tell you where they see the problem areas. Get them to point you to where to find what you need. The key to being accepted by the employees is to listen to them and their ideas. Do not make suggestions to them or try to dictate to them. They work with this stuff every day, they know where the problems are, they just have trouble getting anyone to do anything about them. If you act like you are the only experts - here to tell them how to do their jobs, they will fight you tooth and nail.
The single most critical thing you can do to gain trust is to listen to people and take what they have to say seriously.
Never under any circumstance only talk to managers. Managers do not know the process, but they think they do. The people who actually do the work know alot of details about the processes that the managers won't think to tell you. Some of those details are critical.
Now the really hard part of gaining the trust is that they will be looking to see if what you are doing means they will lose their jobs. It's especially tough if that is exactly what it means. So you need to start having a track record for actually improving things not making them worse before many of the employees will open up to you about the real problems. So start with some low-hanging fruit and get some accomplishments under the team belt. Make sure these things are things that fix long-standing issues and especially are things that do not involve reducing personnel (even if you later intend to do so).