Contacting Tumblr is your best bet for tracking someone down.
That being said, it seems that the general consensus from sources on children's internet safety is that Tumblr is not for children and that there is little that can be done to make it more kid-friendly (it's not as easy to restrict access as, say, Instagram). The ability to message a user anonymously (as sounds like was done in this case) is definitely one of the reasons why. One good source for application-specific guides for parents is yoursphere (I'm not affiliated with it at all).
This article, Tumblr - Why Kids Love It and What Parents Need To Know, is particularly enlightening- summing things up by saying:
The fact of the matter is, Tumblr, like many websites, blog platforms
and social networks, does not have the content filters or oversight in
place to ensure a healthy experience for your child.