Delphi can access many types of databases. Using forms and reports that you create, the BDE (Borland Database Engine) can access local databases, like Paradox and DBase, network SQL server databases, like InterBase, and SysBase, and any data source accessible though ODBC (open database connectivity). On the other hand the BDE also has a number of weakness. Its underlying architecture is oriented toward Paradox and dBASE file server based tables. The BDE is a layer in between your application and the underlying database; such layers always have some kind of performance penalty.
Native MS Access BDE replacement
Take a look at some native third-party DB components that will help you access MS Access databases with Delphi without the BDE. All the components compile directly in the Delphi exe file, you need not to distribute any DLLs.

Top Native MS Access BDE replacement Component Sets
ADO and Delphi
Take a look at some native third-party ADO components that will help you access databases with Delphi without the BDE. Delphi's support for ADO will undoubtedly increase with subsequent Delphi versions, most of the components listed here allow you to use ADO with Delphi versions 3 and up.

Top ADO Delphi VCL Component Sets