There are a handful of potential answers to that question, but there are a few things we need to clear up first.
MySQL hasn't been Open Source since at least August 2007. There is a thread
(click here to view) that discusses the Open Source issues that just opened today. At least for the next few months this shouldn't be an issue, there is very little difference between the Open Source version of MySQL and the version that Sun Microsystems now owns.
There are a number of ways to move a database from Microsoft Access using the Jet database engine to other front ends accessing other database products. Many of these combinations are Open Source.
If your intent is to move primarily to the web, I'd suggest that you just "bite the bullet" up front and convert to PHP. This is not a lot harder than other options, and it is by far the most scalable and the easiest to put up as a web site. It also allows you to retain a great deal of database independance.
If your intent is primarily to move to Open Source but you really intend to keep this a desktop based project, then Open Office is probably a good choice. It is conceptually very much like Microsoft Access and Jet combination.
There are many choices between these two extremes. In order to help you pick one of them, I'd need to know a lot more about what you have done, what you can do now, and what you want to do in the future.
Maybe someone else on the forum can give you a simple answer, but I can't do that beause I don't know enough about what your abilities and needs are.
-PatP