Hi Dana,
I have not seen or heard anything to suggest that
Hollywood & Vine will be closed at the beginning of June 2017 but after checking the Walt Disney World website and attempting to make an Advance Dining Reservation at this particular dining location for dates at the beginning of June 2017, can see the difficulty which you are having.
I have found myself that sometimes, for whatever reason, there appears to be a slight delay in the tables being loaded onto the online dining reservation system for certain restaurants at the Walt Disney World Resort.
I would therefore recommend that you call and speak to a Cast Member on (407) 939-1947 to see if they are able to make an Advance Dining Reservation for you. If they are not, they may be able to provide you with the reason for this and give you some idea of when you will be likely able to make a reservation.
Wishing you a magical trip filled with memories which will last a lifetime!
Emma