To minimize the footprint, I suggest the following :
Use the Microsoft provider for Oracle (System.Data.OracleClient), which ships with the framework.
Download the Oracle Instant Client Package - Basic Lite : this is a zip file with (almost) the bare minimum. I recommend version 10.2.0.4, which is much smaller than version 11.1.0.6.0.
Unzip the following files in a specific folder :
v10 :
oci.dll
orannzsbb10.dll
oraociicus10.dll
v11 :
oci.dll
orannzsbb11.dll
oraociei11.dll
On a x86 platform, add the CRT DLL for Visual Studio 2003 (msvcr71.dll) to this folder, as Oracle guys forgot to read this...
Add this folder to the PATH environment variable.
Use the Easy Connect Naming method in your application to get rid of the infamous TNSNAMES.ORA configuration file. It looks like this : sales-server:1521/sales.us.acme.com.
This amounts to about 19Mb (v10).
If you do not care about sharing this folder between several applications, an alternative would be to ship the above mentioned DLLs along with your application binaries, and skip the PATH setting step.
If you absolutely need to use the Oracle provider (Oracle.DataAccess), you will need :
ODP .NET 11.1.0.6.20 (the first version which allegedly works with Instant Client).
Instant Client 11.1.0.6.0, obviously.