Honestly, if you aren't planning on making a regular thing of it, MDF is probably a better choice. It's more stable than even multi-ply, and cheaper too. The ply wins on edge and corner durability though. On the third hand, MDF from e.g. Home Depot is pretty crap. If you're already going to a better lumberyard, maybe the ply is worth the additional dollars now that you've gone to the trouble of finding a better lumberyard.
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.