One power supply on a digital isolator
Depends on your definition of "work"

If all are interested in is signal continuity then yes, a "digital isolator" would work. However it would no longer be isolating and acting more like a buffer. If all you are after is buffering then yes this will work but there are cheaper alternatives.

An isolated buffer is used when you need to break a ground loop or the reference points are different. 

You are after mitigating a failcase where the relay breaks over. In this case powering both sides of the digital isolator from the same power source will not provide isolation.

Two options


Provide a floating/isolated PSU for the secondary side (https://power.murata.com/mev1s2405sc.html is an example)
Use a digital isolator with integrated power ( http://www.ti.com/product/ISOW7821 )


NOTE: Whatever approach you go for, ensure that the output drive current capability is good enough for the solid-state relay ( 25 mA ). If you look at the isolators listed they are only really viable for ~ 4-10mA output drive capability as they are meant for signal isolation. 

Personally I would go for a SIL DC:DC and utilise an OPTOCOUPLER to provide the 2nd layer of isolation, if I was concerned about this failure mode