RE: No valid Oracle Client found. Please note that TOAD only supports 32 bit Oracle Client 'nstallat'ons.Please view the release notes for additional system requirements

You say you’re using Toad for Oracle 10g – there is no such thing. So I’m guessing you’re running Toad 10.0, 10.1 or 10.5 – all of which are 32-bit only apps
and require 32-bit Oracle client (read the 1st blog below). Toad does not offer a 64-bit version – which you must download specifically – until Toad 11.6.

http://www.toadworld.com/products/toad-for-oracle/b/weblog/archive/2012/11/28/running-toad-for-oracle-in-a-64-bit-environment-updated.aspx

http://www.toadworld.com/products/toad-for-oracle/b/weblog/archive/2012/01/25/64-bit-toad-fact-vs-fiction.aspx