Is it possible for someone not to have their name displayed on the property appraisers website?

It is possible, but difficult to do. It is PUBLIC information, even though others think it is not. You would have to prove that it is somehow dangerous for others to know where you live ie your a state court judge. Even police officers and other elected officials are on that site as well. Now, if you can't convince them to take you off, you can change the ownership -ie put the property into some type of trust - alot of expense for what you are trying to do.