For stability I would go with an expensive multi-ply 1/4" plywood. Not your big-box plywood, I'm talking the high-end cabinet-maker type of plywood (Euro-ply, Apple Ply, etc.).
Further considerations:
Multi-ply is frequently available in 5'x5' sheets and can be tough to find in 4'x8's. MDF is universally 49"x97" (so you can net a 4'x8' after accounting for the dinged up edges). The size of your puzzle may inform your choice.
Home Depot et al. will usually cut a sheet to size on their panel saw. Your local lumberyard may or may not be interested in doing so for a small order.
Home Depot et al. generally aren't interested in delivering for an even half reasonable price if they do it at all. A real lumberyard might be willing to do so, though it's unlikely to be cheap for a single sheet order ($25 for an order under $500, last I needed a delivery. Cheaper than owning a truck or van for me). They also usually dispatch trucks on a schedule. Mine goes north on Tuesdays and south on Thursdays, for example. Plan ahead and don't expect your stuff to show up like Amazon packages the next day.
For the 9.000s I always used paper for the backside (old calendars are a good source for big sheets with excellent quality). Using one wooden lath at the top was enough to ensure stability for a 18.000. The 9.000s didn't need that - just paper and a bit of duct-tape on the backside were enough.