Hi,
The only requirement to access an asp.net web application (if not web service) from client PC is a browser like IE, Firefox etc. You don't need to install Oracle client at client PCs. If asp.net application and Oracle database are running different servers, you need to install Oracle client in the server where the asp.net application is running.
If the application is a desktop application (Windows Forms), you need to install Oracle client at client PC along with installation of that application.