MS Access doesn't play well with
any version of Java. Its biggest problem (outside of its limited data capacity) is that it's not designed to be used in a multi-user environment such as a web application. And the only
JDBC driver that I know of for it is the lowest-common-denominator ODBC-JDBC bridge driver. which not only doesn't play well in the multi-user space but isn't known for being high performance.
Since the preferred way to talk to SQL relational databases is JDBC, you can be pretty comfortable with any of them. MySQL/MariaDB is perhaps the easiest one to find general public support on, but most JDBC-compatible DBMSs are easy to talk to in Java.
Among the ones I've done JBC with are:
Oracle
IBM DB2*
MySQL/MariaDB*
PostgreSQL*
sqlite**
Microsoft SQL Server
Apache Derby*
The starred ones are live production servers I deal with everyday. SQLite tends to be more of an undercover thing, but it's the embedded DBMS in Android devices as well as a popular "invisible" DBMS for Linux apps. For example, I have a recipe filer system that runs SQLite undercover.
Most of the "real" DBMS's require a continuously-running stand-alone database server listening on tcp/ip ports. But not all. SQLite is typically embedded within the app. Apache Derby can be run either way.