The attendee left their briefcase in the auditorium.
Unless you are talking about one attendee leaving a briefcase that belongs to more than one person, this sentence is grammatically incorrect.
Let’s assume that we are talking about one attendee. So, then, you should say:
The attendee left his briefcase in the auditorium.
Using the term ‘their’ implies more than one attendee, so ‘his’ is the correct pronoun choice.
Now you’ve gone and done another bad thing. You have assumed that the attendee was a man! Women have briefcases too, you know!
So now, to be politically correct (unless it was an all-male meeting), you must write:
The attendee left his or her briefcase in the auditorium.
(This sentence is grammatically correct.)
If you don’t like the sound of ‘his or her,’ re-structure your sentence. How about:
One of the attendees left a briefcase in the auditorium.
An attendee left a briefcase in the auditorium.
Remember: Even if you don’t care to be a Grammar Geek or Goddess, you still need to use the correct pronoun reference. The GGs of the world thank you.