You've to get the Oracle Instant Client 32bit and make sure oci.dll & ociw32.dll are in environment variable %PATH%, to make the 32bit DSN work.
But Microsoft ODBC driver for Oracle is based on older client library(OCI7). We would recommend you to use Oracle Instant Client ODBC 32bit for 32bit ODBC DSN.
You can find the installation instruction in the following page:
Success! This was way more complicated than it should have been...downloading two different packages, unzipping them into the same directory, updating the PATH variable, creating a new TNS_ADMIN variable, creating the TNSNAMES.ORA file, and lastly creating the new System DSN and testing it successfully. I do appreciate your assistance!