Can you link to docs in the old system? If so, create Google docs that are organized links into the old system. You want to see the minutes from all the meetings over the last year? Here's that page of links. Budgets? Here it is.
Over time you'll make the Google Docs the de factor standard. Once everyone is accustomed to using Google docs, you can start creating new docs in any system. Including Google docs.
This will gradually wean people off the old system without any single, massive switch. And hopefully it'll be a nice, gradual process.
btw, if the old system doesn't support links into documents, you can ask Bob to add it.