Client computers does not need Access to be installed. .Net libraries support many database providers. See:
.NET Framework Data Providers[
^]
To be able to use MS Access as a database, you need to use OLEDB drivers. That's all.
Note: please, carefully read Sergey's answer. There is a lot of usefull information. You should know that Access database has several limits, such as database size (2GB). You have many alternatives: PostgreSQL, MySQL, SQL Server Express edition, etc. Choose the best one for you (depending on your/application needs/requirements).