Can 22 AWG Solid Copper carry 24 VAC?
Wire carries current, insulation prevents leakage of voltage.

Any wire can carry any voltage up to the working voltage of the insulation. Category 5 phone cable is rated for 125wV and commonly carries 12-57VDC for Power Over Ethernet, 48VDC for phone and 90VAC for phone ringer operation.

Wire size determines the ampacity (amount of current) the wire can carry. As long as your device pulls 600mA or less, 22 gauge wire should be sufficient. If it needs current in the 2 amp range, you'll need 18 gauge wire. Excessive current heats the wire and can melt the insulation.

Another thing to think about is voltage drop and that is affected by the size of the wire (resistance). You will have a 2V drop in available voltage at the end of a 100ft 22ga cable at 24V/600mA. 1V drop for 50ft at 24V/600mA.

Their specs aren't technical enough to tell you what the current draw is on the device.