The thickness of the parts is largely determined by your bend.
Bent lamination thickness.
As to ply thickness.
For very small shaker boxes in cherry even 1 16 can be too thick even after steaming.
Start with a desired thickness thin until it survives the bend radius.
Once the parts are formed changing the overall thickness will be very difficult so take the time to figure out the number and thickness of strips needed to make up your part.
Lamination ply radius thickness 2 4 3 32 4 8 1 8 8 12 3 16 12 1 4.
Bent lamination is a process of bending thin strips of wood using lots of glue and lots of clamps.
The process starts with ripping some thin strips of wood and applying glue to each strip.
Cutting laminates on the bandsaw cutting your own laminates is best done with a bandsaw.
Bottom line do as previously suggested.
For a 6 radius bend using kiln dried walnut the best i could do was 13 laminations per inch.
Shallow curves can be made up of fewer thicker laminations while tighter curves need more thin laminations.
Bending plywood is available in 5 16 and thicknesses and comes in several species including poplar birch and lauan in standard sized sheets 4 x 8 with the veneer layers running either lengthwise or widthwise.
1 8 1 4 is common but i usually end up somewhere around 3 16.
Anything thicker was prone to splitting.