It could work, but:
1) The cities with hospitals are quite far away from each other so it will take a while for Dracula to get to all three of it.
2) The Hunters can (and they should) draw Item cards in the hospital before they move out.
But it is enough if Dracula does it with the closest hospital. For example if he is in Eastern Europe with Hunters on his tail then it is enough to set up Budapest that way, the other two hospitals do not matter anyway.
The hospital is a distinctly different location that the city it is attached to, so I don't think you are allowed to supply at the hospital. You must move to the city and then you may supply.