Door skins are made to repair doors. They are very thin plywood, about 1/8 inch, with a good veneer surface. They are slightly larger than 36 inch by slight longer than a typical 6 foot 8 inch door. Any good lumberyard that does a pro trade will either have them or can order them for you. They are relatively easy to work with, far more so than working with veneer, and for what ever reason are cheaper than the same area of veneer.

Years ago I refaced all the ugly, dark luan mahogany doors in my house with oak door skins. I got it down to about an hour per door using contact cement and using a laminate trimmer to trim the edges and plunge route the passage lock hole. I veneered the jambs and replaced all the casings and stops with solid oak. When done, it was just about impossible to tell that they were not original. OK, I am a nut, but it was easier than replacing all the wallpapers in the house that abutted the doors.

Anyway, given the manner in which your panels are fastened in, it would be a piece of cake to replace with a similar piece of virgin oak plywood or doorskins.

Another thought, since the panels are easily removeable, is to cover them with a commercial quality vinyl wall covering. You would only need to clean and scuff sand the panel, apply a pre-wallpapering bonding agent and hang the wallcovering directly on the old panels with a heavy, pre-mixed "clay paste", available at any good paint store.
Such commercial grade wallcoverings usually come in 54 inch widths and in hundreds of patterns and faux finishes, such as leatherette etc. This route would probably cost under $200 in materials and require almost no special tools, just a wide blade spackle knife for smoothing and some single edge razor blades. Roll the paste on with a small trim roller.

Just some possible alternatives.