In the UK I would go to a builders merchant that also supplies timber and sheet products such as Travis Perkins or a timber merchant and there are companies that just supply board products. I don't think you will find veneered sheets that thin in DIY shops like B & Q or Wickes. You need to check the thickness of the sheet which I think is 4 or 6mm.
I have only been a regular visitor to Spain for the last 5 years and apartment owner for 2 years so am not that familiar with the market. I would start by looking in Leroy Merlin and The Depot just to check that it is not the same as our DIY shops. Then I would look to see if their builders merchants or timber merchants stocked them. If these turned up nothing I would then try and find a Carpenter/Joiner as you often see them on the Industrial Estates (Poligono Industrial) and see if they could supply the item.
I don't think the plywood will be that expensive, maybe €30-40, and it is an easy job to replace it.
I am out there again in two weeks so I will have a look around and see what I can find. Whilst writing this I have just remembered that I have seen similiar metal door and frame sets in Leroy Merlin and the Depot so it could be likely that they would stock the thin plywood veneered sheets to cover them.