They are funded under fire department.
As far as I can tell there are no workers assigned to shelters, if there are workers the game does not show us. As for keeping the shelters stocked I do just assume the food expires, but it really seems that they are set up as part of deliveries same as commercial buildings, even though they are part of the fire department. So basically, if your traffic becomes an issue then they don't receive regular deliveries.
Shelters in real life store food that last for many months like canned and dried/powered items...so they wouldn't need regular deliveries, but hey, it's all about the challenge for the game right?
Anyway, as I stated in an earlier post, I just don't place shelters anymore since they don't seem to do much good and the cost and ability to maintain them just does not seem worth it. The majority of the time they never fill up before the tornado or other disaster hits, if they get hit directly they lose people and I've not noticed much in the way of saved lives with or without them.
( it would make sense to me that many homes would have basements or personal shelters as many do who live in tornado prone areas of the US...but I'm doubtful that is programmed in the game) The disaster relief depots do the most good and I still place weather radar and warning systems so I know where they will strike, I also like the radio towers, but since I have no shelters they are just for ascetics.
I am also a bit dismayed that the high-rise buildings get leveled by tornadoes in the game, it's very doubtful that this would happen, even the devastating EF5 tornado that hit Joplin, Missouri in 2011 didn't level the 8-story hospital...it just blew out all the windows. And the few others that have hit downtown areas of cities have never lost a skyscraper...mainly just the windows.