Degrees of Shakerness
With Shaker furniture, you first need to decide what level of authenticity you are trying to achieve. The Shaker's didn't use plywood, but that doesn't mean that you can't. Also, I generally think of lighter woods for Shaker furniture (cherry especially). The Shaker chest of drawers I built for my son is cherry with no plywood. I even built the drawer bottoms from solid frame and panel poplar and maple. That said, I'd probably never do it again. It was too much work. I'd use solid wood for the outside and plywood for the drawer bottoms. Authentic is nice, but I have a limited lifespan.
Using plywood for the casework is much faster and cheaper, but it will look like plywood, not solid wood. If you decide to use plywood, I wouldn't use MDF. It's just too heavy and I can't imagine the price is low enough once you put a walnut of cherry veneer on it to make a difference.