Standard, commercially available veneers are usually sold in 10 mil sheets. For 3/8" thick, you're talking a stack of 30-35 veneers (depending on lamination technique and thickness of the adhesive). You can get about 36 pcs the size you need from one 4'x8' sheet of veneer. You'd therefore need two sheets.... the cheapest is clear pine at $55 retail... that's $110. If you want to go that route, woodcraft sells it.

Suggest you go the resawing route if you don't want to spend that kind of money. You'd need less than 1 boardfoot... at $3-5/boardfoot, that seems the way to go.

I make drums that are approx. 3/8" out of 3 sheets of ash that are about 1/8" thick each. I resaw to about 3/16" - 1/4" thick, then plane to 1/8". These are easily bent with steam or hot pipe methods.

Doorskins are still sold, they are not single layer veneer tho but 1/8" 3 ply plywood. They are readily available at most lumberyards in luan or birch in 4'x8' sheets. You could laminate this material but you would not get the solid wood appearance on the edges resawn stock would give. The Hardwood Store has 1/8" thick solid stock listed in various woods.
I really doubt anyone would have enough veneer to make a 3/8" thick lamination just laying around, but you never know.

Ron,
The difference between doing bent wood lamination with random pieces vs resawn material is visually dramatic. It depends on the effect you want as to which way to do it.
With resawn material carefully cut from the same billet (done properly using a wide thin resaw blade on the bandsaw, a correctly designed and constructed resaw fence, thickness planed to the appropriate thickness for the amount of bend and wood species, with minimum material loss, and kept in order) and finally glued up in a clamping jig that produces tight even or adjustable pressure will produce bent stock that is difficult to distinguish from solid stock without very close examination. That is the goal of most woodworkers doing the technique.
Bent wood lamination made fom random stock, even when done with strips of the same species, will always give the apperance of plywood on the edges due to even slight color variations from piece to piece, grain direction and pattern changes. Density and grain direction will aso affect the bending properties of each piece making it difficult to get a solid tight glue up without resorting to a massive glue jig and dozens of high quality clamps even for a relatively simple piece. Using overly thin material (veneer) has it's own set of complications that will make it an excercise in masochism, and still not likely give a much better appearance. Don't believe me? Try glue laminating a well used deck of playing cards into a solid curved block. If you manage to get them glued together then bandsaw them in two and examine the interior to see how you did. Playing cards don't have much grain direction so will be much easier to laminate than wood. Expand that difficulty exponentially to the size of your project to get an idea of what you are attempting to do.
If you don't care if you get a solid appearance on the edges of the stock then laminate it out of thin plywood (with a thicker solid facing that is bent or sawn to shape if you want to sculpt the top surface) You could even use a product called "bending plywood" that is designed to be easily curved if you need to make tight bends.
Chair arms, since the curves are shallow, are more often made from selected stock with the appropriate curved shape grain growth. Early wood ship builders learned to select bent trees for ships ribs.
What you can't do with wood is bend it into compound curves, a shape that is bent in two or more directions across the same area. A bowl is the simplest example of a compound curve.