How to use a solid state relay in a dip package to drive a 24 vac relay
A couple of things.

1) Your 24 Vac contactor requires how much current?  You don't say.  But my past experience tells me that the coil on the contactor uses probably about 20 VA.  20VA / 24V = 0.83A.  That means that your SSR is too small for that contactor.

Also be aware that contactors with AC coils have significant inrush current.  If the datasheet for the contactor doesn't give you the inrush current for the coil, you will have to measure it yourself.  Pick your SSR to have a peak current rating higher than the inrush current for your contactor.

2) From the datasheet, the LED in the SSR needs at least 5mA.  5V - 1.5V = 3.5V / 5 mA = 700 Ohms.  Your 560 Ohm resistor is a good value to use.

In general, you are on the right path.  You do need to find a different SSR to use for your contactors but there are suitable parts available.

[Edit]

I mention that contactors with AC coils have significant inrush current.  The datasheet for the contactor will probably use terms like "sealed current" and "un-sealed current".  The "sealed current" is the current that the coil draws when it is fully engaged (the armature is "sealed" to the stator coil).

The "un-sealed" current is what the contactor coil draws when the armature is fully extended from the stator coil.  This current is significantly higher than when the armature is sealed.

If your contactor datasheet doesn't give you the un-sealed current rating or if you don't have access to the datasheet because this is already-existing equipment, you can easily measure the un-sealed current yourself.  Simply block the armature so that it can't pull into the coil and apply power.  Be sure to block it right at the top end of its' travel - the current begins to drop as soon as the armature starts to enter into the coil.

But be really quick when doing this - the coil will burn up if you leave power applied too long while the armature isn't sealed.  It shouldn't take more than a few seconds to measure the current and the coil won't be harmed at all - so long as this does take only a few seconds.