Track how data is inserted into a 3rd party software with data in sql? Is this possible?

So, I've been asked to write some sql that would pull data from one database and insert it into some 3rd party software.  YIKES.   Insert SQL statements aren't the problem,   I'm curious to know if there is a way to trace how/when data is inserted via the 3rd party software that is a sql database.  Let me explain a little more, when a user types in all the information on the UI in the 3rd party software and clicks save, is there a way to track how the sql data is inserted.  Curious what tables are inserted too, in what order, etc..etc.. 
If I could see how the 3rd party software saves the data, I think that would be a good place to start to figure out how I"m going to insert into this sql.  Any advice, suggestions are definately appreciated.
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