PVC foam for this outdoor sign?

HDU or 1" routed PVC would work. We use more PVC than HDU BUT to each his own. I would routed down about 3/16" or 1/4" and it should be just fine

 

Exterior grade PVC works fine outside. I'd buy a sheet of 1" and a sheet of 1/2". Cut the shape out of 1" and texture it (if desired). Cut the raised border and letters out of 1/2". Paint and then assemble.

 

That sign is most appropriately done with 2" HDU foam without a doubt

 

I would route pvc and pocket the letters just 1/16 deeper then route the letters out of 12 mil pvc you could also have the other parts done this way and then paint the parts and inset the painted stuff.