what to use to update our Microsoft Access database
What does this mean? My peers seem to think it refers mainly to updating Access. I might be reading too much into the rest of your post regarding integrating things (maybe MS Project, Quickbooks, etc.). I don't see a need to update what you already have as far as that goes since 20 year old tables aren't going to be any better than those created in the latest version. Besides, you seem to be using 2010 already, so I'm not sure we're getting the full drift. Perhaps the DAO and other libraries in the old version are a problem?
The more we hear silence, the more we begin to think about our value in this universe.
Paraphrase of Professor Brian Cox.