Grant Permission to Execute Set Transaction Isolation Level
First a quick point .. It's pretty unusual someone "needs" to use read uncommitted unless the need is to randomly get bad data.  

Lecture done, permissions are generally found under individual commands.  In this case you are looking at SET TRANSACTION ISOLATION LEVEL.  

Typically you would find a Security/Permissions section, although in this case there isn't one.  I believe that is because this is a command that anyone can run (although some isolation levels have to be enabled).  Another place you can look for permissions is under the properties of the database or server.  You will see a Permissions tab that will have a list of available permissions.  But again I didn't find anything related to the command.

I'm mostly stumped by the fact that you granted db_owner and they worked.  My guess is that it isn't the SET ISOLATION LEVEL that is actually causing the issue.  Can you please include the actual error you are getting?