Allow anonymous user to search custom account field
This can be covered by the Views module in a few steps. 


Make sure anonymous users have the Access User Profiles permission. 
Create a user view with a page display in the Views UI.
Add a exposed filter on your view for your license number field. Set the Operator  to Is equal to if you want a search by full account number only, or set it to Contains if you want a partial/full account number search. 
Adjust the display how you see fit. The display can contain links to the user profile page. 
Save the View. 


From this point you should now have a page from which to search for user profiles with your custom account number field.