TIP120 Transistor for AC voltage?

No the TIP120 can not switch AC. The TOP223Y device is not meant to switch AC either. It is a AC/DC converter switching regulator.

There are several ways to switch AC using DC control voltages with the most often used being the good old mechanical relay. In

your application this would probably be the best choice also. Mechanical relays will require some kind of driver circuit to handle the

coil current and a reverse bias diode across the relay coil pins to handle the inductive voltage spike when the coil is turned off.

Electrically isolating the Pi outputs from the driver circuit with opto-isolators is always a good idea. You can find reasonably priced

already built relay boards for Raspberry Pi by Googling for them. The sprinkler controller uses what is considered to be low voltage

AC for safety reasons (around water it is very important) and this should eliminate most concerns that occur when connecting

these boards to high voltage AC such as normal house line voltage.

You could use TRIACs or back-to-back SCRs.

You can use a transformer to generate your 24VAC.  This will provide your isolation from the line.  Using optical or other types of isolation to protect your controller is a very good idea.  A properly designed solid-state switch will have better MTBF than a relay.  You can use an AC SSR if this is a one-off and you don't want to shoulder much design effort.  You can use TRIAC output optocouplers to have an easy time controlling your thrystors

You could use optically isolated triacs to provide isolation as well as ac switching....then it is just like driving an LED.

Y'know you can switch AC with a MOSFET if you have a supervoltage.

If you really, really, really want to use the TIP120, you could place it in a diode bridge - the diodes would steer the ac so that the transistor only saw dc. (Use an optocoupler to get around all the problems of biassing the transistor appropriately.) But it's a daft thing to do: the bridge and the transistor would dissipate so much power they'd need to be bolted to a heatsink and the volts drop would be so high that the valve might struggle a bit to pull in.

To echo the wise advice above, if it were me I'd either buy a relay board (which is the easy, simply-connect-it-together approach if you just want to get the job done) or view it as a way to get familiar with triacs (if you want a project you can learn from and develop further into other areas). It looks like you'd get some expert advice here if you wanted to go the triac route and being 24Vac it would be much safer than experimenting with mains voltage.

The circuit that John posts in the last comment on this thread gives you a general idea of the circuit you'd be putting together

https://www.element14.com/community/people/jw0752/blog/2016/05/17/analyzing-and-hacking-a-power-supply-board

"A properly designed solid-state switch will have better MTBF than a relay."

Given the amount of use a relay will see in a lawn sprinkler system (once a day worst case, more likely a few times a week)

I don't see this as a problem at all.

Using a little transformer confers a benefit because output power will be limited by the size of the core.  These transformers are ubiquitous in the HVAC industry, IIRC

Even when it is not switching the relay has environmental sensitivity due to potential oxidation of the contacts.  I am not a big fan of electro-mechanical relays, but they have their uses.  Once upon a time mercury-wetted relays were very popular for switching large inductive loads

Bistable relays are really neato where they are indicated

I forgot about that FWBR 'trick.'  Not a bad idea. One could use Shottkys, maybe, to mitigate the drop.