(A) that supplies for the new wing of the hospital, which hospital administrators hoped to open by the end of the year, will be - Misplaced Modifier modifying hospital instead of its wing
(B) that supplies for the hospital's new wing, which hospital administrators hope to be opened by the end of the year, be - Opened is a red flag after "be" form..
(C) that the hospital's new wing's supplies, which hospital administrators hope to be opened by year's end, might be- Misplaced Modifier modifying supplies instead of wing
(D) that supplies for the new wing of the hospital, which hospital administrators hope to be open by the end of the year, were - "were" is red flag due to tense problem
(E) that supplies for the hospital's new wing, which hospital administrators hope to open by the end of the year, be - It correctly uses modifier
So, E