I have been helping to sort out the house of a woman who has died. Among her papers were letters from Queen Elizabeth and Prince Philip, of course written, not by them personally, but by their secretaries. However the letters bear the appropriate letterhead and indicate that the Queen and the Prince know of the matter which is being addressed. 

There is no suggestion that the Palace might refuse to respond because it would interrupt the relationship the letter-writer should have with more local authorities. You write to the Queen, you get a response from the Queen's office.

If the Queen of the Commonwealth of Nations and her consort can reply to every one of the probably thousands of letters they receive, why can't those who sit in the offices of the Vatican?