The one in the picture is a Grumman bodied truck. They used a GM chassis with Grumman's aluminum body atop it.
I can't tell from the pictures but it looks like the wheel bolt pattern is 5/10 lug. That 5 on the front and 10 on the rear. If so, it used a heavier chassis than the pickups did, with a solid beam front axle and leaf springs on all 4 corners.
If its an 8 lug truck, then it used normal SLA independent front suspension carried over from the trucks.
I offer this answer as a person who worked for the Postal Service for a long time doing fleet maintenance and repair on those trucks.