Attach a trigger to 'sys.dm_db_missing_index_details'
No triggers on system DMVs, sorry.

Besides, you wouldn't really want to do this: that DMV is updated every time a query is run with a missing index request. That could be hundreds or thousands of times per second - and performance matters there.

Instead, consider running an Agent job every 5-15 minutes to capture the data you're looking for.