TCAS uses the transponder system. Transponder off? TCAS inop.
Transponder "off" is another condition than transponder "standy".
Standby of a transponder is defined in the Mode S transponder MOPS as:
Standby: Means of selecting the condition in which all transponder functions, other than transmissions on the reply frequency and associated self-testing, are operational (i.e., the Standby condition).
I am not 100% sure how Standby would affect TCAS, because I don't have the TCAS technical documents here with me. Basically TCAS works by interrogating on the interrogation frequency of 1030 MHz and listening to replies on the reply frequency of 1090 MHz.
In Standby mode, a transponder will not reply on 1090 MHz, and therefore it will not be detected by TCAS systems in other aircraft or ATC radar. However, I don't think Standby of a transponder necessarily inhibits transmission on 1030 MHz so your TCAS might still interrogate other aircraft and listen to replies.