Replication is almost certainly the way to go, although MySQL Replication is generally only one-way only unless you use a cluster, which is a little out of my comfort zone to talk about.
You could have a MySQL database which is accessible to your clients and also via a data source name via the access database. That way both update the same database, just with 2 different views.
Unless you go for a complicated MySQL cluster replication solution, remember that it’s only one way updates, so one will have to be the master copy and the other a slave (or replicated) copy, possibly used for heavy reporting or internal read only accessing.
In any case, if you internet connection is flakey then it’s highly likely that the replication will not function at it’s best and your local database and the online database will need to be communication with each other at any time when a change it made. It’s the flakey internet connection that will make this tricky and if that is taken out of the equation, it’s a simple master and slave scenario, one I use myself daily.
Link to Official MySQL 5.0 Replication.