How to perform same DML statements on two different servers (Oracle and SQL-Server) simultaneously
I do not have enough reputation to write a comment so I am writing here. It may be an unswer to your question.

I have never tried, but it should work. You can add an Oracle database into your SQL Server as a linked server and then write a trigger in the sql server and perform the insert to Oracle via OpenQuery. This should be the easiest way and should work.