https://store.nintendo.com/ng3/repair/repairStatusUnRegisteredUser.jsp
I'm not exactly sure what the reference number might be, it could be a "Order or Repair Number" that you use at the above url to check the status of your repair.
Or it could be used when you contact Nintendo again and they can use that reference number to more easily find your information that is logged on their computer systems.
And yes, you can set up an online repair (more information at the above url), though it still might be easier to just call back Nintendo to discuss this with the service representative, so that both your Wii U console and GamePad are logged onto just one repair order, and that nothing is accidentally missed that needs repairing.
And also a very big yes, the service representatives are very nice and patient, as they are also gamers too and only want to get you back to fully enjoying your Nintendo products as fast as possible.