Hi Eddie,
Great suggestion! We do not currently support this but we do show the eBay user ID on the orders page (seen HERE).
We're always looking for ways to build the Global Search tool up and have even recently added some new search functionality to search by tracking, order number, alternate order number, names, and email addresses. It certainly seems that including the option to search by user ID would be another fitting enhancement.
I've gone ahead and forwarded your request to our product team. Please keep an eye on this thread for any future updates.