Looking for a 24vac switch from 3v inputs

This isn't too hard to implement. I can see the box and controller getting out of sync, but if the controller hits the zone twice and it doesn't matter what order the two sub-zones come on in, then that seems fine.

You are right in that you need some kind of memory. Since the unit will have no power to it between uses, that memory needs to be non-volatile. A microcontroller with built in EEPROM would do fine. EEPROMs are only good for a finite number of writes, but that's 100s of 1000s at least so no issue there.

When the power to the switch box turns on, all it really does is run the micro. The micro then turns on one of N relays to route the power to one of the sub-zones. It also writes the new state to its EEPROM so that it will power the next sub-zone in sequence next time.

A tiny micro running at slow clock speed can easily handle this. The 5V current will be small so a linear regulator will do well enough and be simple. Get relays that can run directly from the full wave rectified AC so there are no power conversion issues. 24V AC after full wave bridge with filter cap should be around 30-32 Volts. "24V" DC relays would work but get a little warm. Genuine 30V relays may be harder to find, so you could get 24V relays and put a resistor in series with the coil. A reverse catch diode accross the coil and a NPN transistor with base resistor to the micro is all you need per sub-zone output.

Another thing to consider is that the micro needs to see one power up each time the main controller turns on the zone. This should be as simple as putting a little low pass filtering on the micro's reset input so that it doesn't start running until a 100 ms or so after power is applied. By that time glitches and switching transients should be over.

The main controller also needs to leave some off time between powering this zone so that it toggles to the next sub-zone. It will take some time for the voltage to drop before the micro loses power or is shut down by the reset circuit. It could be a second or two depending on what values are chosen.

The more I think about it, the more I'm realizing the trickiest part of this is the reset circuit. You want to make sure the micro runs cleanly once per power up, and that it goes into reset cleanly once on power down and not too long after power down. This is all quite doable, but something that needs to be considered.

Vee should be connected to Ground, or to a negative supply.

As this is a CMOS part, all digital inputs MUST be connected (possibly via a resistor) to either Vcc or Ground at all times - otherwise the inputs may float to a "maybe" state, with unknown consequences.

Connect all S0 - S2 and /E to ground with 10K (or so) resistors. Also connect A0 to ground with a 10K resistor, and A1 to Vcc.

Now, the output should initially be Low. Pulling S0 up to Vcc will select A1, and the output should be High.