Only if the local time zone of the client is actually relevant, you should store it in a TIMESTAMP WITH TIME ZONE column.
INSERT INTO PRCLOG (LOGDTE, LOGSID, LOGPRC, LOGSEQ, LOGTXT) VALUES (CURRENT_TIMESTAMP AT TIME ZONE '+00:00', sys_context('userenv','sid'), SUBSTR(v_caller, 1, 128), gLogIndex, SUBSTR(v_msg, 1, 4000));
You can issue ALTER SESSION SET TIME_ZONE from a stored procedure using EXECUTE IMMEDIATE.
I you want your stored time to be automatically adjusted to the session time zone, then you need to use TIMESTAMP WITH LOCAL TIME ZONE and make sure the database time zone is '+00:00' (i.e. UTC/GMT). Plane TIMESTAMP data type does not do any calculation by itself. It is like a DATE data type except that fractional seconds are available.