From what I recall, we've seen "food slots" in the standard Rec Rooms (numerous episodes, such as
Naked Time), the Arboretum Rec Room (
And the Children Shall Lead) and the Transporter Room (
Tomorrow is Yesterday).
As
T'Girl has said, there's nothing to say that any of these rooms weren't immediately adjacent to (either vertically or horizontally) the food preparation room. FJ's notion of mini-dumbwaiters snaking through the ship is now quite unnecessary. Any food slots in the Secondary Hull can be managed by a second food prep room (which would also be essential in the event of saucer separation!)
The only thing that points to something other than a simple human operated galley and dumbwaiter system is the speed of the service. In
Naked Time, Tormolen puts the food card in and has his plate of steaming food within 3 seconds, barely enough time for the plate to be put onto the tray! Clearly there must be some sort of mechanical aid in operation. Bearing in mind all the problems with TOS era personnel transporters, perhaps the food slots use a cruder but more reliable type for non-living subjects? This would jive with McCoy's line in
Arena about "reconstituted meals", because clearly they don't taste right!
Finally, Kirk's cabin was also seen to have a food slot in
Mirror, Mirror (and remained afterwards), although it was smaller and only ever dispensed drinks (and then only once) so I don't count that in the same category - it's probably just a 23rd century drinks cabinet
