SQL Server is a complete separate product than Access (although the two can, and often, work together), and MySQL is a web-based database. So, no, you need neither... Access has it's own databse engine (ACE, formerly JET) which serves more or less the same purpose that SQLS and MySQL do (if in a much more lightweight fasion than either of those two).
So, put your Access Backend on the server share, and that's all you need to do in regards to setting up the data end of the DB. Then, link your tables to that Access BE from the frontends. (this is done automatically if you use the wizard).
It's very important to make sure that each user gets a separate copy of the FE though... under no circumstances should more than one person access the same FE file at once.
Stop back with any other questions.
Cheers,