The permissions are those of the user executing the code, so whoever updates the table has their permissions used.

I doubt most users have access to sys.dm_exec_connections. I would suggest you use a trace instead of a trigger to trap what is happening. You can set up a server side trace to get this.

Depending on what you are looking for, you could also use the default trace to watch activity (http://www.sqlservercentral.com/articles/64547/)