A family of Microsoft relational database management systems designed for ease of use.
Running Access over a VPN is never a good idea and is known to be a potential cause of corruption.
SharePoint is one option, but personally wouldn't go that route because of the list limitations. If you are going to migrate,. Which makes a lot of sense after this issue existing for more than 30+ months now, I'd be more looking at SQL Server (Express) or SQL Azure, or another RDMS. True Enterprise grade database, which SharePoint is not. Keep a database, a database!
SQL Server (Express) -> Local Instances
SQL Azure -> Web, if you're looking to share with users all over the place