Found the issue and thought I would post her so people would not wonder.

The answer turned out to be simple. I was thinking about the reply by Jeff Moden and his comment about there being a long running query and decided to take another look at the code involved. It turns out there was no index on the "report" table that included the trackingnumber field. This is the field that the triggers in the attached file use to join to the "report" table to update the correct record.

I added a non-clustered index, just on that field for now, since I was skeptical that would have any effect. I mean, come on! Deadlock troubleshooting is supposed to be so much more complex and the solution supposed to be more complicated than this.

Nope. Those particular deadlocks just went away and the application became much more responsive in the related area.

Thanks, Jeff.