I'm with Phil, probably your best bet is a trigger. Here are the docs. You want to look at the DML triggers and, probably, an AFTER trigger, meaning, after the insert/update/delete, not before.
An alternative would be to create an Extended Event session. You'd have to identify all procedures & functions that call the table in question and add them as filtered objects to rpc_completed. You'd also need to catch sql_batch_completed and filter based on the table name.
A trigger in this case is going to be a lot easier to implement. However, I suspect, you may see a lighter load on the system with the Extended Event. Testing is your friend in this, and all cases.