Water cooler faucet handle broke
I've not found an image to match the handles in the photo. Without a model designation, it's challenging to zero in on the same product, but the images I've found indicate there are similarities in your product and the parts available. 

My search terms were "sunbeam water cooler handles" and could be modified by adding your model number for best results. I was surprised to see that WalMart carries replacement parts, but that's likely to be online and not necessarily available in stores. Amazon also popped up in the search results.



It appears that you may be able to dismantle the nozzle just above the opening if parts are available for that segment, or you should be able to remove the entire assembly for replacement as well.