Can one connect to Toad for Oracle using ODBC?

NO.  Toad for Oracle requires SQL Net client and does a direct connection – end of story.

People often ask this question due to a huge misunderstanding about ODBC. They often think that ODBC somehow replaces the need for SQL Net on their client PC’s. It does not. Even if you want to use ODBC drivers to create an Oracle database connection – you must still have SQL Net. ODBC is just an API. Microsoft provides the upper portion of the API call interface – which is not database centric (works for all db’s). But you still must have the lower portion of the API call interface – and that’s provided by each database vendor. For Oracle, that means you still must have SQL Net – even if you are using ODBC.