There is a difference between creating an application using Access and using an Access database to simply store data that is accessed by an application written in a diffferent language.
If you create an application that uses forms, reports, queries, etc., then you MUST have Access installed on the target machine to run it. If you do not wish to purchase full retail versions then you can purchase the Office Developers edition which will allow you to distribute the runtime version of Access along with your application database to an unlimited number of client machines.
If your application is written in some other language and you are simply using ADO to access the data, then all you need is Jet which is normally installed with Office. If you don't have Jet installed, you should be able to download it from the Microsoft download site.